What Is a Modified Cholesky Factorization?

Newton methods for minimizing a function F:\mathbb{R}^n \to \mathbb{R} generate a sequence of points x_k, where the step from x_k to x_{k+1} is along a search direction p_k determined from a linear system G_k p_k = -g_k, where g_k =  \nabla F(x_k) is the gradient and G_k is an approximation to the Hessian matrix \nabla^2 F(x_k). The equation g_k^Tp_k = - p_k^T G_k p_k shows that G_k is a descent direction if p_k^T G_k p_k > 0, and in order to guarantee that this condition holds for all p_k we need G_k to be positive definite. But even if G_k is the exact Hessian, positive definiteness is not guaranteed far from a minimizer. We can modify the method to ensure definiteness of G_k, as with quasi-Newton methods. Or we can perturb the matrix, if necessary, to make it positive definite. Modified Cholesky factorization perturbs and factorizes the matrix at the same time. It is useful in other situations, too, such as in constructing preconditioners and in bounding the distance to a positive semidefinite matrix.

A modified Cholesky factorization of a symmetric matrix A is a factorization P(A + E)P^T = LDL^T, where P is a permutation matrix, L is unit lower triangular, and D is diagonal or block diagonal and positive definite. It follows that A+E is a positive definite matrix.

A natural way to compute a modified Cholesky factorization is to modify the Cholesky factorization algorithm. Cholesky factorization fails when it tries to take the square root of a negative quantity or divide by zero. We can avoid both possibilities by increasing nonpositive pivots when they are encountered. This corresponds to making a diagonal perturbation E to A and computing a Cholesky factorization A + E = R^T\!R. However, choosing a suitable E is more difficult than it might seem.

Consider the matrix

\notag   A =   \begin{bmatrix}    1 & 1          & 1    & 0 \\    1 & 1-\epsilon & 2  & 1\\    1 & 2          & 1    & 1\\    0 & 1          & 1    & 1   \end{bmatrix}, \quad 0 < \epsilon \ll 1.

Since Cholesky factorization generates the same sequence of Schur complements as Gaussian elimination, it suffices to consider Gaussian elimination. The diagonal elements of R are the square roots of the pivots. After one step of elimination the reduced matrix is

\notag   A^{(2)} =   \left[\begin{array}{c|ccc}    1 & 1          & 1 & 1\\\hline    0 & -\epsilon &  1   & 1\\    0 &  1          & 0 & 1\\    0 &   1         & 1 & 1   \end{array}\right],

and the trailing 3\times 3 matrix (a Schur complement) is clearly indefinite because the (2,2) entry, which is the next pivot, is negative. We can make the (2,2) entry positive by adding 2\epsilon to it:

\notag   A^{(2)} + E =   \left[\begin{array}{c|ccc}    1 & 1          & 1 & 1\\\hline    0 & \epsilon & 1   & 1\\    0 & 1           & 0 & 1\\    0 & 1           & 1 & 1   \end{array}\right]   \quad   (E =  2 \mskip1mu\epsilon \mskip1mu e_2e_2^T).

The next stage of the factorization can complete and it yields

\notag   A^{(3)} =   \left[\begin{array}{cc|cc}    1 & 1          & 1 & 1 \\    0 & \epsilon   & 1 & 1 \\\hline\rule{0cm}{18pt}    0 & 0          & -\displaystyle\frac{1}{\epsilon} & 1 - \displaystyle\frac{1}{\epsilon}\\\rule{0cm}{20pt}    0 & 0          &1 - \displaystyle\frac{1}{\epsilon} &  1 - \displaystyle\frac{1}{\epsilon}   \end{array}\right],

The trailing 2\times 2 submatrix has elements of order \epsilon^{-1} \gg 1. Not only will a perturbation of order \epsilon^{-1} be required to the (3,3) element to allow the Cholesky factorization to continue, but the Cholesky factor will have elements of order \epsilon^{-1/2} so numerical stability will likely be lost. Yet the smallest eigenvalue of A is of order 1, so it should have been possible to make only an O(1) perturbation to A in order for the factorization to succeed.

This example shows that if we are to increase a pivot element then we need a more sophisticated strategy that takes account of the size of the resulting elements of the factors and the effect on later stages of the factorization.

A modified Cholesky factorization should satisfy, as far as possible, four objectives.

  • If A is “sufficiently positive definite” then E is zero.
  • If A is indefinite, \|E\| is not much larger than

    \notag       \min \{\, \|\Delta A\| : A+\Delta A~ \text{is positive semidefinite} \,\}

    for some appropriate norm.

  • The matrix A+E is reasonably well conditioned.
  • The cost of the algorithm is the same as the cost of standard Cholesky factorization, that is, n^3/3 + O(n^2) flops for an n\times n matrix.

Gill and Murray (1974) gave the first modified Cholesky algorithm, which computes P(A + E)P^T = LDL^T with diagonal D and E. It was refined by Gill, Murray, and Wright in 1981. Schnabel and Eskow (1990) gave an algorithm that attempts to produce smaller values of \|E\|, partly by exploiting eigenvalue bounds obtained from Gershgorin’s theorem. That algorithm was subsequently improved by Schnabel and Eskow (1999).

A different approach was taken by Cheng and Higham (1998), building on an earlier idea by Bunch and Sorensen. This approach computes a block \mathrm{LDL^T} factorization PAP^T = LDL^T, were P is a permutation matrix, L is unit lower triangular, and D is block diagonal with diagonal blocks of size 1 or 2. The pivoting strategy is the symmetric rook pivoting strategy of Ashcraft, Grimes, and Lewis (1998), which has the key property of producing a bounded L factor. The cost of pivoting is typically O(n^2) comparisons but can be as large as O(n^3) in the worst case. Cheng and Higham compute the perturbation \Delta D of minimal Frobenius norm such that D + \Delta D has eigenvalues greater than or equal to \delta, where \delta > 0 is a parameter. The modified Cholesky factorization is then P(A+E)P^T = L(D+\Delta D)L^T.

A significant advantage of the block \mathrm{LDL^T} approach is that it is modular: any implementation of the factorization can be used and the modification is simply inexpensive postprocessing of the D factor. The other algorithms are not simple modifications of an \mathrm{LDL^T} factorization and are not straightforward to implement in an efficient way as a block algorithm. Note that in the block \mathrm{LDL^T} approach E is a full matrix and it is not explicitly computed.

Modified Cholesky software is not widely available in libraries. Implementations of the Cheng–Higham algorithm are available in

Example

We take the 4\times 4 matrix above with \epsilon = 10^{-2}:

\notag   A =   \begin{bmatrix}    1 & 1          & 1    & 0 \\    1 & 0.99       & 2  & 1\\    1 & 2          & 1    & 1\\    0 & 1          & 1    & 1   \end{bmatrix}.

It has eigenvalues

-1.0050e+00  -2.3744e-01   1.0000e+00   4.2325e+00

The Gill–Murray–Wright algorithm computes as E the diagonal matrix with diagonal elements

0   2.0200e+00   2.0000e+00            0

while the Schnabel–Eskow algorithm (1999) computes E with diagonal elements

1.0000e+00   1.0050e+00   1.0050e+00   1.0000e+00

For the Cheng–Higham algorithm with \delta = (2u)^{1/2} \|A\|_F = 6.7 \times10^{-8} (where u \approx 1.11 \times 10^{-16} is the unit roundoff), the perturbed matrix A+E is

1.0000e+00   1.0000e+00   1.0000e+00            0
1.0000e+00   1.4950e+00   1.4975e+00   9.9749e-01
1.0000e+00   1.4975e+00   1.5000e+00   1.0025e+00
         0   9.9749e-01   1.0025e+00   2.0100e+00

The Frobenius norms of the perturbations to A are 2.84, 2.00, and 1.43, respectively, and the 2-norm condition numbers are 33.8, 43.2, and 4.67 \times 10^8. The large condition number for the Cheng–Higham algorithm is caused by the value of the parameter \delta. With \delta = 0.1, the perturbed matrix is

1.0000e+00   1.0000e+00   1.0000e+00            0
1.0000e+00   1.5453e+00   1.4475e+00   9.9724e-01
1.0000e+00   1.4475e+00   1.5497e+00   1.0027e+00
         0   9.9724e-01   1.0027e+00   2.1100e+00

at Frobenius norm distance 1.57 from A and with 2-norm condition number 327.3. For comparison, the symmetric matrix with all eigenvalues greater than or equal to 0.1 that is closest to A in the Frobenius norm is at a distance 1.15 from A.

In general, there is no clear ordering of the different modified Cholesky methods in terms of their ability to satisfy the four criteria.

References

This is a minimal set of references, which contain further useful references within.

Related Blog Posts

This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.

What Is the Matrix Sign Function?

The matrix sign function is the matrix function corresponding to the scalar function of a complex variable

\notag      \mathrm{sign}(z) =      \begin{cases}          1, & \mathop{\mathrm{Re}} z > 0, \\         -1, & \mathop{\mathrm{Re}} z < 0.      \end{cases}

Note that this function is undefined on the imaginary axis. The matrix sign function can be obtained from the Jordan canonical form definition of a matrix function: if A = XJX^{-1} is a Jordan decomposition with \mathrm{diag}(J) = \mathrm{diag}(\lambda_i) then

\notag     S = \mathrm{sign}(A) = X \mathrm{sign}(J)X^{-1}                          = X \mathrm{diag}(\mathrm{sign}(\lambda_i))X^{-1},

since all the derivatives of the sign function are zero. The eigenvalues of S are therefore all \pm 1. Moreover, S^2 = I, so S is an involutory matrix.

The matrix sign function was introduced by Roberts in 1971 as a tool for model reduction and for solving Lyapunov and algebraic Riccati equations. The fundamental property that Roberts employed is that (I+S)/2 and (I-S)/2 are projectors onto the invariant subspaces associated with the eigenvalues of A in the open right-half plane and open left-half plane, respectively. Indeed without loss of generality we can assume that the eigenvalues of A are ordered so that J = \mathrm{diag}(J_1,J_2), with the eigenvalues of J_1\in\mathbb{C}^{p\times p} in the open left half-plane and those of J_2\in\mathbb{C}^{q\times q} in the open right half-plane (p + q = n). Then

\notag     S = X \begin{bmatrix} -I_p & 0 \\                            0   & I_q           \end{bmatrix}X^{-1}

and, writing X = [X_1~X_2], where X_1 is n\times p and X_2 is n\times q, we have

\notag \begin{aligned}   \displaystyle\frac{I+S}{2} &= X \begin{bmatrix} 0 & 0 \\                                                0   & I_q           \end{bmatrix}X^{-1} = X_2 X^{-1}(p+1\colon n,:),\\[\smallskipamount]   \displaystyle\frac{I-S}{2} &= X \begin{bmatrix} I_p & 0 \\                                                  0   & 0           \end{bmatrix}X^{-1} = X_1 X^{-1}(1\colon p,:). \end{aligned}

Also worth noting are the integral representation

\notag  \mathrm{sign}(A) = \displaystyle\frac{2}{\pi}                     A \int_0^{\infty} (t^2I+A^2)^{-1} \,\mathrm{d}t

and the concise formula

\notag   \mathrm{sign}(A) = A (A^2)^{-1/2}.

Application to Sylvester Equation

To see how the matrix sign function can be used, consider the Sylvester equation

\notag       AX - XB = C, \quad A \in \mathbb{C}^{m\times m}, \; B \in                    \mathbb{C}^{n\times n}, \;                     C \in \mathbb{C}^{m\times n}.

This equation is the (1,2) block of the equation

\notag          \begin{bmatrix} A & -C \\                    0 & B \end{bmatrix}            =          \begin{bmatrix} I_m & X   \\                    0   & I_n \end{bmatrix}          \begin{bmatrix} A & 0 \\                    0 & B \end{bmatrix}          \begin{bmatrix} I_m & X   \\                    0   & I_n \end{bmatrix}^{-1}.

If \mathrm{sign}(A) = I and \mathrm{sign}(B) = -I then

\notag    \mathrm{sign} \left(          \begin{bmatrix} A & -C \\                    0 & -B \end{bmatrix}          \right)          =          \begin{bmatrix} I_m &   X \\                    0   & I_n \end{bmatrix}          \begin{bmatrix} I_m & 0    \\                    0   & -I_n \end{bmatrix}          \begin{bmatrix} I_m & -X   \\                    0   & I_n \end{bmatrix} =          \begin{bmatrix} I_m & -2X   \\                    0   &  -I_n \end{bmatrix},

so the solution X can be read from the (1,2) block of the sign of the block upper triangular matrix \bigl[\begin{smallmatrix}A & -C\\ 0& B \end{smallmatrix}\bigr]. The conditions that \mathrm{sign}(A) and \mathrm{sign}(B) are identity matrices are satisfied for the Lyapunov equation AX + XA^* = C when A is positive stable, that is, when the eigenvalues of A lie in the open right half-plane.

A generalization of this argument shows that the matrix sign function can be used to solve the algebraic Riccati equation XFX - A^*X - XA - G = 0, where F and G are Hermitian.

Application to the Eigenvalue Problem

It is easy to see that S = \mathrm{sign}(A) satisfies \mathrm{trace}(S) = \mathrm{trace}(\mathrm{diag}(\mathrm{sign}(\lambda_i))) = q - p, where p and q are the number of eigenvalues in the open left-half plane and open right-half plane, respectively (as above). Since n = p + q, we have the formulas

\notag    p = \displaystyle\frac{n - \mathrm{trace}(S)}{2}, \quad    q = \displaystyle\frac{n + \mathrm{trace}(S)}{2}.

More generally, for real a and b with a < b,

\notag   \displaystyle\frac{1}{2} \bigl( \mathrm{sign}(A - aI) - \mathrm{sign}(A - bI) \bigr)

is the number of eigenvalues lying in the vertical strip \{\, z: \mathop{\mathrm{Re}}z \in(a,b)\,\}. Formulas also exist to count the number of eigenvalues in rectangles and more complicated regions.

Computing the Matrix Sign Function

What makes the matrix sign function so interesting and useful is that it can be computed directly without first computing eigenvalues or eigenvectore of A. Roberts noted that the iteration

\notag     X_{k+1} = \displaystyle\frac{1}{2} (X_k + X_k^{-1}), \quad X_0 = A,

converges quadratically to \mathrm{sign}(A). This iteration is Newton’s method applied to the equation X^2 = I, with starting matrix A. It is one of the rare circumstances in which explicitly inverting matrices is justified!

Various other iterations are available for computing \mathrm{sign}(A). A matrix multiplication-based iteration is the Newton–Schulz iteration

\notag     X_{k+1} = \displaystyle\frac{1}{2} X_k (3I - X_k^2), \quad X_0 = A.

This iteration is quadratically convergent if \|I-A^2\| < 1 for some subordinate matrix norm. The Newton–Schulz iteration is the [1/0] member of a Padé family of rational iterations

X_{k+1} = X_k  p_{\ell m}^{}\left(1-X_k^2\right)  q_{\ell m}^{}\left(1-X_k^2\right)^{-1},             \quad X_0 = A,

where p_{\ell m}(\xi)/q_{\ell m}(\xi) is the [\ell/m] Padé approximant to (1-\xi)^{-1/2} (p_{\ell m} and q_{\ell m} are polynomials of degrees at most \ell and m, respectively). The iteration is globally convergent to \mathrm{sign}(A) for \ell = m-1 and \ell = m, and for \ell \ge m+1 it converges when \|I-A^2\| < 1, with order of convergence \ell+m+1 in all cases.

Although the rate of convergence of these iterations is at least quadratic, and hence asymptotically fast, it can be slow initially. Indeed for n = 1, if |a| \gg 1 then the Newton iteration computes x_1 = (a+1/a)/2 \approx a/2, and so the early iterations make slow progress towards \pm 1. Fortunately, it is possible to speed up convergence with the use of scale parameters. The Newton iteration can be replaced by

\notag   X_{k+1} = \displaystyle\frac{1}{2} \bigl(\mu_kX_k + \mu_k^{-1}X_k^{-1}\bigr), \quad X_0 = A,

with, for example,

\notag     \mu_k = \sqrt{\|X_k^{-1}\|/\|X_k\|}.

This parameter \mu_k can be computed at no extra cost.

As an example, we took A = gallery('lotkin',4), which has eigenvalues 1.887, -1.980\times10^{-1}, -1.228\times10^{-2}, and -1.441\times10^{-4} to four significant figures. After six iterations of the unscaled Newton iteration X_6 had an eigenvalue -100.8, showing that X_6 is far from \mathrm{sign}(A), which has eigenvalues \pm 1. Yet when scaled by \mu_k (using the 1-norm), after six iterations all the eigenvalues of X were within distance 10^{-16} of \pm 1, and the iteration had converged to within this tolerance.

The Matrix Computation Toolbox contains a MATLAB function signm that computes the matrix sign function. It computes a Schur decomposition then obtains the sign of the triangular Schur factor by a finite recurrence. This function is too expensive for use in applications, but is reliable and is useful for experimentation.

Relation to Matrix Square Root and Polar Decomposition

The matrix sign function is closely connected with the matrix square root and the polar decomposition. This can be seen through the relations

\notag     \mathrm{sign}\left( \begin{bmatrix} 0 & A \\\ I & 0 \end{bmatrix} \right )      = \begin{bmatrix}0 & A^{1/2} \\ A^{-1/2} & 0 \end{bmatrix}, \\[\smallskipamount]

for A with no eigenvalues on the nonpositive real axis, and

\notag    \mathrm{sign}\left( \begin{bmatrix} 0 & A \\\ A^* & 0 \end{bmatrix} \right )     = \begin{bmatrix}0 & U \\ U^* & 0 \end{bmatrix},

for nonsingular A, where A = UH is a polar decomposition. Among other things, these relations yield iterations for A^{1/2} and U by applying the iterations above to the relevant block 2n\times 2n matrix and reading off the (1,2) block.

References

This is a minimal set of references, which contain further useful references within.

Related Blog Posts

This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.

What Is a Fractional Matrix Power?

A pth root of an n\times n matrix A is a matrix X such that X^p = A, and it can be written X = A^{1/p}. For a rational number r = j/k (where j and k are integers), defining A^r is more difficult: is it (A^j)^{1/k} or (A^{1/k})^j? These two possibilities can be different even for n = 1. More generally, how can we define A^\alpha for an arbitrary real number \alpha?

Recall, first, that for a nonzero complex scalar z we define z^\alpha = \mathrm{e}^{\alpha\log z}, where \log is the principal logarithm: the one taking values in the strip \{\, z : -\pi <  \mathop{\mathrm{Im}} z \le \pi \,\}. We can generalize this definition to matrices. For a nonsingular matrix A we define

\notag        A^\alpha = \mathrm{e}^{\alpha\log A}.

Here the logarithm is the principal matrix logarithm, the matrix function built on the principal scalar logarithm, and so the eigenvalues of \log A lie in the strip \{\, z : -\pi <  \mathop{\mathrm{Im}} z \le \pi \,\}. When \alpha \in [-1,1], the eigenvalues of A^\alpha, which are \lambda^\alpha where \lambda is an eigenvalue of A, lie in the segment \{\, z: -|\alpha|\pi < \arg z \le |\alpha|\pi\,\} of the complex plane.

The most important special case is \alpha = 1/p for a positive integer p, in which case

\notag        A^{1/p} = \mathrm{e}^{\frac{1}{p}\log A}.

We can check that

\notag    \bigl(A^{1/p}\bigr)^p = \bigl(\mathrm{e}^{\frac{1}{p}\log A}\bigr)^p                          = \mathrm{e}^{\log A} = A,

so the definition does indeed produce a pth root. The matrix A^{1/p} is called the principal pth root.

Returning to the case of rational p, we note that

\notag    A^{j/k} = \mathrm{e}^{\frac{j}{k}\log A}            = \Bigl(\mathrm{e}^{\frac{1}{k}\log A}\Bigr)^j            = (A^{1/k})^j,

but (A^{j})^{1/k} can be a different matrix. In general, it is not true that (A^\alpha)^\beta = (A^\beta)^\alpha for real \alpha and \beta, although for symmetric positive definite matrices this identity does hold because the eigenvalues are real and positive.

An integral expression for A^\alpha valid for \alpha \in(0,1) is

\notag    A^\alpha  = \displaystyle\frac{\sin(\alpha\pi)} {\alpha\pi} A \int_0^{\infty}(t^{1/\alpha}I+A)^{-1}\,\mathrm{d}t,    \quad \alpha \in(0,1).

Another representation for A = t (I - B) with \rho(B) < 1 is given by the binomial expansion

\notag     A^{\alpha} =  t^{\alpha}          \displaystyle\sum_{j=0}^\infty {\alpha \choose j}(-B)^j,     \quad \rho(B) < 1.

For 2\times 2 real matrices of the form

\notag A =  \begin{bmatrix}        a & b \\        c & a      \end{bmatrix}

with bc < 0 there is an explicit formula for A^\alpha. It is easy to see that A has eigenvalues \lambda_{\pm} = a \pm \mathrm{i} d, where d = (-bc)^{1/2}. Let \theta = \arg(\lambda_+) \in (0,\pi) and r = |\lambda_+|. It can be shown that

\notag A^\alpha = \displaystyle\frac{r^\alpha}{d}        \begin{bmatrix}            d\cos(\alpha\theta) & b\sin(\alpha\theta) \\            c\sin(\alpha\theta) & d\cos(\alpha\theta)        \end{bmatrix}.

Computation

The formula A^\alpha = \mathrm{e}^{\alpha \log A} can be used computationally, but it is somewhat indirect in that one must approximate both the exponential and the logarithm. A more direct algorithm based on the Schur decomposition and Padé approximation of the power function is developed by Higham and Lin (2013). MATLAB code is available from MathWorks File Exchange.

If A is diagonalizable, so that A = XDX^{-1} for some nonsingular X with D = \mathrm{diag}(\lambda_i), then A^\alpha = X D^\alpha X^{-1} = X \mathrm{diag}(\lambda_i^\alpha) X^{-1}. This formula is safe to use computationally only if X is well conditioned. For defective (non-diagonalizable) A we can express A^\alpha in terms of the Jordan canonical form, but this expression is not useful computationally because the Jordan canonical form cannot be reliably computed.

Inverse Function

If X = A^{1/2} then A = X^2, by the definition of square root. If X = A^{\alpha} does it follow that A = X^{1/\alpha}? Clearly, the answer is “no” in general because, for example, X = A^2 does not imply A = X^{1/2}.

Using the matrix unwinding function it can be shown that (A^{\alpha})^{1/\alpha} = A for \alpha \in [-1,1]. Hence the function g(A) = A^{1/\alpha} is the inverse function of f(A) = A^{\alpha} for \alpha\in[-1,1].

Backward Error

How can we check the quality of an approximation X to A^{\alpha}? For \alpha = 1/p we can check the residual A - X^p, but for real p there is no natural residual. Instead we can look at the backward error.

For an approximation X to A^\alpha, a backward error is a matrix \Delta A such that X = (A + \Delta A)^\alpha. Assume that A and X are nonsingular and that \alpha \in [-1,1]. Then, as shown in the previous section, X = (A + \Delta A)^\alpha implies \Delta A = X^{1/\alpha} - A. Hence the normwise relative backward error is

\notag      \eta(X) = \displaystyle\frac{ \|X^{1/\alpha} - A \|}{\|A\|}, \quad \alpha\in[-1,1].

Applications with Stochastic Matrices

An important application of fractional matrix powers is in discrete-time Markov chains, which arise in areas including finance and medicine. A transition matrix for a Markov process is a matrix whose (i,j) element is the probability of moving from state i to state j over a time step. It has nonnegative entries and the rows sum to 1, so it is a stochastic matrix. In practice, a transition matrix may be estimated for a certain time period, say one year, but a transition matrix for a shorter period, say one month, may be needed. If A is a transition matrix for a time period p then a stochastic pth root of A is a transition matrix for a time period a factor p smaller. Therefore p = 12 (years to months) and p = 7 (weeks to days) are among the values of interest. Unfortunately, A^{1/p} is not necessarily a stochastic matrix. Moreover, A can have a stochastic pth root that is not A^{1/p}. For example, the stochastic matrix

\notag    A = \left[\begin{array}{ccc}        0  & 1 & 0\\        0  & 0 & 1\\        1  & 0 & 0\\        \end{array} \right]

has principal square root

\notag    A^{1/2} = \frac{1}{3} \left[\begin{array}{rrr}        2  & 2 & -1 \\        -1 & 2 & 2 \\        2 & -1 & 2        \end{array} \right],

but A^{1/2} is not stochastic because of the negative elements. The square root

\notag         Y = \left[\begin{array}{ccc}          0  & 0  &  1\\          1  & 0  &  0\\          0  & 1  &  0\\          \end{array} \right]

is stochastic, though. (Interestingly, A is also a square root of Y!)

A wide variety configurations is possible as regards existence, nature (primary or nonprimary), and number of stochastic roots. Higham and Lin (2011) delineate the various possibilities that can arise. They note that the stochastic lower triangular matrix

\notag    A_n = \begin{bmatrix}             1         &                   &           &    \\      \frac{1}{2}      &  \frac{1}{2}      &           &    \\      \vdots           &  \vdots           &\ddots     &    \\      \frac{1}{n}      &  \frac{1}{n}      &\cdots     &  \frac{1}{n}    \end{bmatrix}

has a stochastic pth root, namely A_n^{1/p}, for all p. For example, to three significant figures,

\notag  A_6^{1/3} = \begin{bmatrix} 1.000  &        &        &        &        &        \\ 0.206  & 0.794  &        &        &        &        \\ 0.106  & 0.201  & 0.693  &        &        &        \\ 0.069  & 0.111  & 0.190  & 0.630  &        &        \\ 0.050  & 0.075  & 0.109  & 0.181  & 0.585  &        \\ 0.039  & 0.056  & 0.076  & 0.107  & 0.172  & 0.550  \\ \end{bmatrix}.

The existence of stochastic roots of stochastic matrices is connected with the embeddability problem, which asks when a nonsingular stochastic matrix A can be written A = \mathrm{e}^Q for some Q with q_{ij} \ge 0 for i\ne j and \sum_{j=1}^n q_{ij} = 0, i=1\colon n. Kingman showed in 1962 that this condition holds if and only if for every positive integer p there exists a stochastic X_p such that A = X_p^p.

Applications in Fractional Differential Equations

Fractional matrix powers arise in the numerical solution of differential equations of fractional order, especially partial differential equations involving fractional Laplace operators. Here, the problem may be one of computing A^\alpha b, in which case for large problems it is preferable to directly approximate A^\alpha b, for example by Krylov methods or quadrature methods, rather than to explicitly compute A^\alpha.

References

This is a minimal set of references, which contain further useful references within.

What Is the Matrix Unwinding Function?

Care is needed when dealing with multivalued functions because identities that hold for positive scalars can fail in the complex plane. For example, it is not always true that (z-1)^{1/2}(z+1)^{1/2} = (z^2-1)^{1/2} or \log(z_1 z_2) = \log z_1 + \log z_2 for all z,z_1,z_2\in\mathbb{C}. Here, the square root is the principal square root (the one lying in the right half-plane) and the logarithm is the principal logarithm (the one with imaginary part in (-\pi,\pi]).

A powerful tool for dealing with multivalued complex functions is the unwinding number, defined for z\in\mathbb{C} by

\notag     \mathcal{U}(z) = \displaystyle\frac{z - \log \mathrm{e}^z}{2\pi \mathrm{i}}.

The unwinding number provides a correction term for the putative identity \log \mathrm{e}^z = z, in that

\notag    \qquad\qquad\qquad\qquad     z = \log \mathrm{e}^z + 2\pi \mathrm{i}\, \mathcal{U}(z)    \qquad\qquad\qquad\qquad (*)

for all z.

A useful formula for the unwinding number is

\notag    \mathcal{U}(z) = \left\lceil\displaystyle\frac{\mathrm{Im} z - \pi}{2\pi}\right\rceil,

where \lceil\cdot\rceil is the ceiling function, which returns the smallest integer greater than or equal to its argument. It follows that \mathcal{U}(z) = 0 if and only if \mathrm{Im} z \in (-\pi, \pi]. Hence \log \mathrm{e}^z = z if and only if \mathrm{Im} z \in (-\pi, \pi].

The unwinding number provides correction terms for various identities. For example, for z_1,z_2\in\mathbb{C}, replacing z by \log z_1 \pm \log z_2 in (*), we have

\notag \begin{aligned}     \log z_1 \pm \log z_2       &= \log \bigl( \mathrm{e}^{\log z_1 \pm \log z_2} \bigr)          + 2\pi\mathrm{i} \, \mathcal(\log z_1 \pm \log z_2)\\       &= \log \bigl(\mathrm{e}^{\log z_1} \mathrm{e}^{\pm \log z_2} \bigr)          + 2\pi\mathrm{i} \, \mathcal(\log z_1 \pm \log z_2)\\       &= \log \bigl( z_1 z_2^{\pm 1} \bigr)          + 2\pi\mathrm{i} \, \mathcal(\log z_1 \pm \log z_2). \end{aligned}

This gives the identities

\notag \begin{aligned} \log (z_1 z_2) &= \log z_1 + \log z_2 - 2\pi \mathrm{i}\, \mathcal{U}(\log z_1 +\log z_2), \\  \qquad\qquad\qquad   \log (z_1/z_2) &= \log z_1 - \log z_2 -       2\pi \mathrm{i}\,\mathcal{U}(\log z_1 - \log z_2).  \qquad\qquad\qquad (\#) \end{aligned}

Note that in textbooks one can find identities such as \log (z_1 z_2) = \log z_1 + \log z_2, in which each occurrence of \log is interpreted as a possibly different branch. For computational purposes we want formulas that contain a specific branch, usually the principal branch.

An application of the unwinding number to matrix functions is in computing the logarithm of a 2\times 2 upper triangular matrix. For any function f, we have

\notag     f\left( \begin{bmatrix}                      \lambda_1 & t_{12}  \\                            0   & \lambda_2             \end{bmatrix} \right)          = \begin{bmatrix}               f(\lambda_1) & t_{12} f[\lambda_1,\lambda_2] \\                 0          & f(\lambda_2)             \end{bmatrix},

where the divided difference

\notag   f[\lambda_1,\lambda_2]    = \begin{cases}      \displaystyle\frac{f(\lambda_2)-f(\lambda_1)}{\lambda_2-\lambda_1},      & \lambda_1 \ne \lambda_2, \\      f'(\lambda_1), & \lambda_1 = \lambda_2.     \end{cases}

When \lambda_1 \approx \lambda_2 this formula suffers from numerical cancellation. For the logarithm, we can rewrite it, using (\#), as

\notag  \begin{aligned}   \log\lambda_2 - \log\lambda_1   &= \log \left(\frac{\lambda_2}{\lambda_1}\right) + 2\pi \mathrm{i} \,        \mathcal{U}(\log \lambda_2 - \log \lambda_1)       \\   &= \log \left(\frac{1+z}{1-z}\right) + 2\pi \mathrm{i}\,     \mathcal{U}(\log \lambda_2 - \log \lambda_1),  \end{aligned}

where z = (\lambda_2-\lambda_1)/(\lambda_2+\lambda_1). Using the hyperbolic arc tangent, defined by

\notag     \mathrm{atanh}(z) = \frac{1}{2}\log\left( \displaystyle\frac{1+z}{1-z} \right),

we obtain

\notag   f[\lambda_1,\lambda_2]    = \displaystyle\frac{2\mathrm{atanh}(z) + 2\pi \mathrm{i}\,      \mathcal{U}(\log \lambda_2 - \log \lambda_1)}{\lambda_2-\lambda_1},    \quad \lambda_1 \ne \lambda_2.

Assuming that we have an accurate \mathrm{atanh} function this formula will provide an accurate value of f[\lambda_1,\lambda_2] provided that z is not too close to \pm 1 (the singularities of \mathrm{atanh}) and not too large. This formula is used in the MATLAB function logm.

Matrix Unwinding Function

The unwinding number leads to the matrix unwinding function, defined for A\in\mathbb{C}^{n\times n} by

\notag   \mathcal{U}(A) = \displaystyle\frac{A - \log \mathrm{e}^A}{2\pi \mathrm{i}}.

Here, \log is the principal matrix logarithm, defined by the property that its eigenvalues have imaginary parts in the interval (-\pi,\pi]. It can be shown that \mathcal{U}(A) = 0 if and only if the imaginary parts of all the eigenvalues of A lie in the interval (-\pi, \pi]. Furthermore, \mathcal{U}(A) is a diagonalizable matrix with integer eigenvalues.

As an example, the matrix

\notag   A = \left[\begin{array}{rrrr}    3 & 1 & -1 & -9\\    -1 & 3 & 9 & -1\\    -1 & -9 & 3 & 1\\    9 & -1 & -1 & 3 \end{array}\right],    \quad \Lambda(A) = \{ 2\pm 8\mathrm{i}, 4 \pm 10\mathrm{i} \}

has unwinding matrix function

\notag   X = \mathcal{U}(A) = \mathrm{i} \left[\begin{array}{rrrr}    0 & -\frac{1}{2} & 0 & \frac{3}{2}\\    \frac{1}{2} & 0 & -\frac{3}{2} & 0\\    0 & \frac{3}{2} & 0 & -\frac{1}{2}\\    -\frac{3}{2} & 0 & \frac{1}{2} & 0 \end{array}\right],    \quad \Lambda(X) = \{ \pm 1, \pm 2 \}.

In general, if A is real then \mathcal{U}(A) is pure imaginary as long as A has no eigenvalue with imaginary part that is an odd multiple of \pi.

The matrix unwinding function is useful for providing correction terms for matrix identities involving multivalued functions. Here are four useful matrix identities, along with cases in which the correction term vanishes. See Aprahamian and Higham (2014) for proofs.

  • For nonsingular A and \alpha,\beta\in\mathbb{C},

    \notag   (A^\alpha)^{\beta} = A^{\alpha\beta}   \mathrm{e}^{-2\beta\pi \mathrm{i}\, \mathcal{U}(\alpha\log A)}.

    If \beta is an integer then the correction term is I. If \alpha\in(-1,1] and \beta = 1/\alpha then \mathcal{U}(\alpha\log A) = 0 and so

    \notag      (A^\alpha)^{1/\alpha} = A, \quad \alpha \in [-1,1],

    and this equation is obviously true for \alpha = -1, too.

  • If A and B are nonsingular and AB = BA then

    \notag      \log(AB^{\pm1}) = \log A \pm \log B - 2\pi \mathrm{i}\, \mathcal{U}(\log A \pm \log B).

    If \arg\lambda_i + \arg\mu_i \in(-\pi,\pi] for every eigenvalue \lambda_i of A and the corresponding eigenvalue \mu_i of B (there is a correspondence because of the commutativity of A and B, which implies that they are simultaneously unitarily diagonalizable), then \log(AB^{\pm 1}) = \log A \pm \log B.

  • If A and B are nonsingular and AB = BA then for any \alpha\in\mathbb{C},

    \notag   (AB)^\alpha = A^\alpha B^\alpha \mathrm{e}^{-2\pi\, \alpha \mathrm{i}\, \mathcal{U}(\log A + \log B)}.

    If \alpha is an integer or the eigenvalues of A and B have arguments in (-\pi/2,\pi/2] then (AB)^\alpha = A^\alpha B^\alpha.

  • For nonsingular A and \alpha\in\mathbb{C}.

    \notag     \log A^{\alpha} = \alpha \log A - 2\pi \mathrm{i}\, \mathcal{U}(\alpha \log A).

    If \alpha\in(-1,1] then \log A^{\alpha} = \alpha \log A, and this equation also holds for \alpha = -1 if A has no negative eigenvalues.

The matrix unwinding function can be computed by an adaptation of the Schur–Parlett algorithm. The algorithm computes a Schur decomposition and re-orders it into a block form with eigenvalues having the same unwinding number in the same diagonal block. The unwinding function of each diagonal block is then a multiple of the identity and the off-diagonal blocks are obtained by the block Parlett recurrence. This approach gives more accurate results than directly evaluating \mathcal{U}(A) from its definition in terms of the matrix logarithm and natrix exponential. A MATLAB code unwindm is available at https://github.com/higham/unwinding

Matrix Argument Reduction

The matrix unwinding function can be of use computationally for reducing the size of the imaginary parts of the eigenvalues of a matrix. The function

\notag   \mathrm{mod}(A) = A - 2\pi\mathrm{i}\, \mathcal{U}(A)

has eigenvalues \lambda with \mathrm{Im} \lambda \in(-\pi,\pi]. Using \mathrm{mod}(A) in place of A can be useful in computing the matrix exponential by the scaling and squaring algorithm because \mathrm{e}^A = \mathrm{e}^{\mathrm{mod}(A)} and \mathrm{mod}(A) can have a much smaller norm than A, giving potential reductions in cost. Combined with the Schur decomposition-based algorithm for \mathcal{U}(A) mentioned above, this idea leads to a numerical method for \mathrm{e}^A. See Aprahamian and Higham (2014) for details.

Round Trip Relations

If you apply a matrix function and then its inverse do you get back to where you started, that is, is f^{-1}(f(z)) = z? In principle yes, but if the inverse is multivalued the answer is not immediate. The matrix unwinding function is useful for analyzing such round trip relations. As an example, if A has no eigenvalues with real part of the form k \pi for an integer k, then

\notag    \mathrm{acos}(\cos A) = \bigl(A-2\pi\,\mathcal{U}    (\mathrm{i} A)\bigr)\mathrm{sign} \bigl(A-2\pi\mathcal{U}( \mathrm{i} A)\bigr).

Here, \mathrm{acos} is the principal arc cosine defined in Aprahamian and Higham (2016), where this result and analogous results for the arc sine, arc hyperbolic cosine, and arc hyperbolic sine are derived; and \mathrm{sign} is the matrix sign function.

History

The unwinding number was introduced by Corless, Hare and Jeffrey in 1996, to help implement computer algebra over the complex numbers. It was generalized to the matrix case by Aprahamian and Higham (2014).

References

This is a minimal set of references, which contain further useful references within.

Related Blog Posts

This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.

What Is a (Non)normal Matrix?

An n\times n matrix is normal if A^*A = AA^*, that is, if A commutes with its conjugate transpose. Although the definition is simple to state, its significance is not immediately obvious.

The definition says that the inner product of the ith and jth columns equals the inner product of the ith and jth rows for all i and j. For i = j this means that the ith row and the ith column have the same 2-norm for all i. This fact can easily be used to show that a normal triangular matrix must be diagonal. It then follows from the Schur decomposition that A is normal if it is unitarily diagonalizable: A = QDQ^* for some unitary Q and diagonal D, where D contains the eigenvalues of A on the diagonal. Thus the normal matrices are those with a complete set of orthonormal eigenvectors.

For a general diagonalizable matrix, A = XDX^{-1}, the condition number \kappa(X) = \|X| \|X^{-1}\| can be arbitrarily large, but for a normal matrix X can be taken to have 2-norm condition number 1. This property makes normal matrices well-behaved for numerical computation.

Many equivalent conditions to A being normal are known: seventy are given by Grone et al. (1987) and a further nineteen are given by Elsner and Ikramov (1998).

The normal matrices include the classes of matrix given in this table:

Real Complex
Diagonal Diagonal
Symmetric Hermitian
Skew-symmetric Skew-Hermitian
Orthogonal Unitary
Circulant Circulant

Circulant matrices are n\times n Toeplitz matrices in which the diagonals wrap around:

\notag      \begin{bmatrix} c_1     & c_n    & \dots   & c_2     \\                      c_2     & c_1    & \dots   & \vdots  \\                      \vdots  & \ddots & \ddots  & c_n     \\                      c_n     & \dots  & c_2     & c_1     \\      \end{bmatrix}.

They are diagonalized by a unitary matrix known as the discrete Fourier transform matrix, which has (r,s) element \exp( -2\pi \mathrm{i} (r-1)(s-1) / n ).

A normal matrix is not necessarily of the form given in the table, even for n = 2. Indeed, a 2\times 2 normal matrix must have one of the forms

\notag    \left[\begin{array}{@{\mskip2mu}rr@{\mskip2mu}}      a & b\\      b & c    \end{array}\right], \quad    \left[\begin{array}{@{}rr@{\mskip2mu}}      a & b\\     -b & a    \end{array}\right].

The first matrix is symmetric. The second matrix is of the form aI + bJ, where J = \bigl[\begin{smallmatrix}\!\phantom{-}0 & 1\\\!-1 & 0 \end{smallmatrix}\bigr] is skew-symmetric and satisfies J^2 = -I, and it has eigenvalues a \pm \mathrm{i}b.

It is natural to ask what the commutator C = AA^*- A^*A can look like when A is not normal. One immediate observation is that C has zero trace, so its eigenvalues sum to zero, implying that C is an indefinite Hermitian matrix if it is not zero. Since an indefinite matrix has at least two different nonzero eigenvalues, C cannot be of rank 1.

In the polar decomposition A = UH, where U is unitary and H is Hermitian positive semidefinite, the polar factors commute if and only if A is normal.

The field of values, also known as the numerical range, is defined for A\in\mathbb{C}^{n\times n} by

F(A) = \biggl\{\, \displaystyle\frac{z^*Az}{z^*z} : 0 \ne z \in \mathbb{C}^n \, \biggr\}.

The set F(A) is compact and convex (a nontrivial property proved by Toeplitz and Hausdorff), and it contains all the eigenvalues of A. Normal matrices have the property that the field of values is the convex hull of the eigenvalues. The next figure illustrates two fields of values, with the eigenvalues plotted as dots. The one on the left is for the nonnormal matrix gallery('smoke',16) and that on the right is for the circulant matrix gallery('circul',x) with x constructed as x = randn(16,1); x = x/norm(x).

fv_ex.jpg

Measures of Nonnormality

How can we measure the degree of nonnormality of a matrix? Let A have the Schur decomposition A = QTQ^*, where Q is unitary and T is upper triangular, and write T = D+M, where D = \mathrm{diag}(\lambda_i) is diagonal with the eigenvalues of A on its diagonal and M is strictly upper triangular. If A is normal then M is zero, so \|M\|_F is a natural measure of how far A is from being normal. While M depends on Q (which is not unique), its Frobenius norm does not, since \|A\|_F^2 = \|T\|_F^2 = \|D\|_F^2 + \|M\|_F^2. Accordingly, Henrici defined the departure from normality by

\notag   \nu(A) = \biggl( \|A\|_F^2 - \displaystyle\sum_{j=1}^n |\lambda_j|^2 \biggr)^{1/2}.

Henrici (1962) derived an upper bound for \nu(A) and Elsner and Paardekooper (1987) derived a lower bound, both based on the commutator:

\notag   \displaystyle\frac{\|A^*A-AA^*\|_F}{4\|A\|_2}   \le \nu(A) \le \Bigl( \displaystyle\frac{n^3-n}{12} \Bigr)^{1/4} \|A^*A-AA^*\|_F^{1/2}.

The distance to normality is

\notag   d(A) = \min \bigl\{\, \|E\|_F: ~A+E \in \mathbb{C}^{n\times n}~~\mathrm{is~                                   normal} \,\bigr\}.

This quantity can be computed by an algorithm of Ruhe (1987). It is trivially bounded above by \nu(A) and is also bounded below by a multiple of it (László, 1994):

\notag     \displaystyle\frac{\nu(A)}{n^{1/2}} \le d(A) \le \nu(A).

Normal matrices are a particular class of diagonalizable matrices. For diagonalizable matrices various bounds are available that depend on the condition number of a diagonalizing transformation. Since such a transformation is not unique, we take a diagonalization A = XDX^{-1}, D = \mathrm{diag}(\lambda_i), with X having minimal 2-norm condition number:

\kappa_2(X) = \min\{\, \kappa_2(Y): A = YDY^{-1}, ~D~\mathrm{diagonal} \,\}.

Here are some examples of such bounds. We denote by \rho(A) the spectral radius of A, the largest absolute value of any eigenvalue of A.

  • By taking norms in the eigenvalue-eigenvector equation Ax = \lambda x we obtain \rho(A) \le \|A\|_2. Taking norms in A = XDX^{-1} gives \|A\|_2 \le \kappa_2(X) \|D\|_2 = \kappa_2(X)\rho(A). Hence

\notag    \displaystyle\frac{\|A\|_2}{\kappa_2(X)} \le \rho(A) \le \|A\|_2.

  • If A has singular values \sigma_1 \ge \sigma_2 \ge \cdots \ge   \sigma_n and its eigenvalues are ordered |\lambda_1| \ge |\lambda_2| \ge \cdots \ge |\lambda_n|, then (Ruhe, 1975)

    \notag      \displaystyle\frac{\sigma_i(A)}{\kappa_2(X)}      \le |\lambda_i(A)|      \le \kappa_2(X) \sigma_i(A), \quad i = 1\colon n.

    Note that for i=1 the previous upper bound is sharper.

  • For any real p > 0,

    \notag    \displaystyle\frac{\rho(A)^p}{\kappa_2(X)} \le     \|A^p\|_2 \le \kappa_2(X) \rho(A)^p.

  • For any function f defined on the spectrum of A,

    \notag    \displaystyle\frac{\max_i|f(\lambda_i)|}{\kappa_2(X)} \le     \|f(A)\|_2 \le \max_i|f(\lambda_i)|.

For normal A we can take X unitary and so all these bounds are equalities. The condition number \kappa_2(X) can therefore be regarded as another measure of non-normality, as quantified by these bounds.

References

This is a minimal set of references, which contain further useful references within.

What Is the Matrix Logarithm?

A logarithm of a square matrix A is a matrix X such that \mathrm{e}^X = A, where \mathrm{e}^X is the matrix exponential. Just as in the scalar case, the matrix logarithm is not unique, since if \mathrm{e}^X = A then \mathrm{e}^{X+2k\pi\mathrm{i}I} = A for any integer k. However, for matrices the nonuniqueness is complicated by the presence of repeated eigenvalues. For example, the matrix

\notag    X(t) =     2\pi \mathrm{i}       \begin{bmatrix}1 & -2t & 2t^2 \\                      0 &  -1 & -t \\                      0 &  0 &  0       \end{bmatrix}

is an upper triangular logarithm of the 3\times 3 identity matrix I_3 for any t, whereas the obvious logarithms are the diagonal matrices 2\pi\mskip1mu\mathrm{i}\mskip1mu\mathrm{diag}(k_1,k_2,k_3), for integers k_1, k_2, and k_3. Notice that the repeated eigenvalue 1 of I_3 has been mapped to the distinct eigenvalues 2\pi, -2\pi, and 0 in X(t). This is characteristic of nonprimary logarithms, and in some applications such strange logarithms may be required—an example is the embeddability problem for Markov chains.

An important question is whether a nonsingular real matrix has a real logarithm. The answer is that it does if and only if the Jordan canonical form contains an even number of Jordan blocks of each size for every negative eigenvalue. This means, in particular, that if A has an unrepeated negative eigenvalue then it does not have a real logarithm. Minus the n\times n identity matrix has a real logarithm for even n but not for odd n. Indeed, for n=2,

\notag    X =  \pi\left[\begin{array}{@{}rr@{}}              0 & 1\\              -1 & 0             \end{array}\right]

satisfies \mathrm{e}^X = -I_2, as does Y = ZXZ^{-1} for any nonsingular Z, since \mathrm{e}^Y = Z\mathrm{e}^X Z^{-1} = -Z Z^{-1}  = -I_2.

For most practical purposes it is the principal logarithm that is of interest, defined for any matrix with no negative real eigenvalues as the unique logarithm whose eigenvalues lie in the strip \{\, z : -\pi < \mathrm{Im}(z)  < \pi \,\}. From this point on we assume that A has no negative eigenvalues and that the logarithm is the principal logarithm.

Various explicit representations of the logarithm are available, including

\notag \begin{aligned}       \log A  &= \int_0^1 (A-I)\bigl[ t(A-I) + I \bigr]^{-1} \,\mathrm{d}t, \\     \log(I+X) &= X - \frac{X^2}{2} + \frac{X^3}{3}                    - \frac{X^4}{4} + \cdots, \quad \rho(X)<1, \end{aligned}

where the spectral radius \rho(X) = \max\{\, |\lambda| : \lambda~\textrm{is an eigenvalue of}~X\,\}. A useful relation is \log (A^{\alpha}) = \alpha \log A for \alpha\in[-1,1], with important special cases \log (A^{-1}) = - \log A and \log (A^{1/2}) = \frac{1}{2} \log A (where the square root is the principal square root). Recurring the latter expression gives, for any positive integer k,

\notag       \log(A) = 2^k \log\bigl(A^{1/2^k}\bigr).

This formula is the basis for the inverse scaling and squaring method for computing the logarithm, which chooses k so that E = I - A^{1/2^k} is small enough that \log(I + E) can be efficiently approximated by Padé approximation. The MATLAB function logm uses the inverse scaling and squaring method together with a Schur decomposition.

References

This references contains more on the facts above, as well as further references.

Related Blog Posts

This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.

What Is a QR Factorization?

A QR factorization of a rectangular matrix A\in\mathbb{R}^{m\times n} with m\ge n is a factorization A = QR with Q\in\mathbb{R}^{m\times m} orthonormal and R\in\mathbb{R}^{m\times n} upper trapezoidal. The R factor has the form R = \left[\begin{smallmatrix}R_1\\ 0\end{smallmatrix}\right], where R_1 is n\times n and upper triangular. Partitioning Q conformably with R we have

\notag    A = QR     =    \begin{array}[b]{@{\mskip-20mu}c@{\mskip0mu}c@{\mskip-1mu}c@{}}    & \mskip10mu\scriptstyle n & \scriptstyle m-n  \\       \mskip15mu          \begin{array}{r}              \scriptstyle m          \end{array}~    &       \multicolumn{2}{c}{\mskip-15mu          \left[\begin{array}{c@{~}c@{~}}                  Q_1 & Q_2                \end{array}\right]       }    \end{array} \mskip-10mu    \begin{array}[b]{@{\mskip-25mu}c@{\mskip-20mu}c@{}}    \scriptstyle n    \\    \multicolumn{1}{c}{        \left[\begin{array}{@{}c@{}}                  R_1\\                  0              \end{array}\right]}    & \mskip-12mu\          \begin{array}{l}              \scriptstyle n \\              \scriptstyle m-n          \end{array}    \end{array}    = Q_1 R_1.

There are therefore two forms of QR factorization:

  • A = QR is the full QR factorization,
  • A = Q_1R_1 is the reduced (also called economy-sized, or thin) QR factorization.

To prove the existence of a QR factorization note that if A has full rank then A^T\!A is symmetric positive definite. Since A = Q_1R_1 implies A^T\!A = R_1^TQ_1^TQ_1R_1 = R_1^TR_1, we can take R_1 to be the Cholesky factor of A^T\!A and then define Q_1 = AR_1^{-1}. The resulting Q_1 has orthonormal columns because

\notag        Q_1^TQ_1 = R_1^{-T} A^T A R_1^{-1}                 = R_1^{-T} R_1^T R_1 R_1^{-1}                  = I.

Therefore when A has full rank there is a unique reduced QR factorization if we require R_1 to have positive diagonal elements. (Without this requirement we can multiply the ith column of Q and the ith row of R by -1 and obtain another QR factorization.)

When A has full rank the columns of Q_1 span the column space (or range) of A. Indeed Ax = Q_1R_1x = Q_1(R_1x) implies \mathrm{range}(A) \subseteq \mathrm{range}(Q_1) while Q_1x = Q_1R_1\cdot R_1^{-1}x =: Ay implies \mathrm{range}(Q_1) \subseteq \mathrm{range}(A), so \mathrm{range}(Q_1) = \mathrm{range}(A). Furthermore, Q^TA = R gives Q_2^TA = 0, so the columns of Q_2 span the null space of A^T.

The QR factorization provides a way of orthonormalizing the columns of a matrix. An alternative is provided by the polar decomposition A = UH, where U has orthonormal columns and H is positive semidefinite. The orthogonal polar factor U is the closest matrix with orthonormal columns to A in any unitarily invariant norm, but it is more expensive to compute than the Q factor.

There are three standard ways of computing a QR factorization.

Gram–Schmidt orthogonalization computes the reduced factorization. It has the disadvantage that in floating-point arithmetic the computed \widehat{Q} is not guaranteed to be orthonormal to the working precision. The modified Gram–Schmidt method (a variation of the classical method) is better behaved numerically in that \|\widehat{Q}^T\widehat{Q} - I\|_F \le c_1(m,n)\kappa_2(A)u for some constant c_1(m,n), where u is the unit roundoff, so the loss of orthogonality is bounded.

Householder QR factorization and Givens QR factorization both construct Q^T as a product of orthogonal matrices that are chosen to reduce A to upper trapezoidal form. In both methods, at the start of the kth stage we have

\notag   \qquad\qquad\qquad\qquad    A^{(k)} = Q_{k-1}^T A =    \begin{array}[b]{@{\mskip35mu}c@{\mskip20mu}c@{\mskip-5mu}c@{}c}    \scriptstyle k-1 &    \scriptstyle 1 &    \scriptstyle n-k &    \\    \multicolumn{3}{c}{    \left[\begin{array}{c@{\mskip10mu}cc}    R_{k-1} & y_k  & B_k \\        0   & z_k  & C_k    \end{array}\right]}    & \mskip-12mu    \begin{array}{c}    \scriptstyle k-1 \\    \scriptstyle m-k+1    \end{array}    \end{array},   \qquad\qquad\qquad\qquad (*)

where R_{k-1} is upper triangular and Q_{k-1} is a product of Householder transformations or Givens rotations. Working on A^{(k)}(k:m,k:n) we now apply a Householder transformation or n-k Givens rotations in order to zero out the last n-k elements of z_k and thereby take the matrix one step closer to upper trapezoidal form.

Householder QR factorization is the method of choice for general matrices, but Givens QR factorization is preferred for structured matrices with a lot of zeros, such as upper Hessenberg matrices and tridiagonal matrices.

Both these methods produce Q in factored form and if the product is explicitly formed they yield a computed \widehat{Q} that is orthogonal to the working precision, that is, \|\widehat{Q}^T\widehat{Q} - I\|_F \le c_2(m,n)u, for some constant c_2.

Modified Gram–Schmidt, Householder QR, and Givens QR all have the property that there exists an exactly orthogonal Q such that the computed \widehat{R} satisfies

\notag   A + \Delta A = Q \widehat{R}, \quad   \|\Delta A\|_F \le c_3(m,n)u \|A\|_F,

for some constant c_3.

Another way of computing a QR factorization is by the technique in the existence proof above, via Cholesky factorization of A^T\!A. This is known as the Cholesky QR algorithm and it has favorable computational cost when m \gg n. In its basic form, this method is not recommended unless A is extremely well conditioned, because the computed \widehat{Q} is far from orthonormal for ill conditioned matrices. The method can be made competitive with the others either by using extra precision or by iterating the process.

Column Pivoting and Rank-Revealing QR Factorization

In practice, we often want to compute a basis for the range of A when A is rank deficient. The basic QR factorization may not do so. Householder QR factorization with column pivoting reveals rank deficiency by incorporating column interchanges. At the kth stage, before applying a Householder transformation to (*), the column of largest 2-norm of C_k, the jth say, is determined, and if its norm exceeds that of z_K then the kth and (k+j)th columns of A^{(k)} are interchanged. The result is a factorization A\Pi = QR, where \Pi is a permutation matrix and R satisfies the inequalities

\notag    |r_{kk}|^2 \ge \displaystyle\sum_{i=k}^j |r_{ij}|^2,     \quad j=k+1\colon n, \quad k=1\colon n.

In particular,

\notag   \qquad\qquad\qquad\qquad\qquad\qquad   |r_{11}| \ge |r_{22}| \ge \cdots \ge |r_{nn}|.   \qquad\qquad\qquad\qquad\qquad\qquad (\dagger)

If A is rank deficient then R has the form

\notag R = \begin{bmatrix}R_{11} & R_{12} \\ 0 & 0 \end{bmatrix},

with R_{11} nonsingular, and the rank of A is the dimension of R_{11}.

Near rank deficiency of A to tends to be revealed by a small trailing diagonal block of R, but this is not guaranteed. Indeed for the Kahan matrix

\notag        U_n(\theta) = \mathrm{diag}(1,s,\dots,s^{n-1})                   \begin{bmatrix} 1 & -c & -c     & \dots & -c \\                               & 1  & -c     & \dots & -c \\                               &    & \ddots &\ddots & \vdots \\                               &    &        &\ddots & -c \\                               &    &        &       &  1                    \end{bmatrix}

where c =\cos\theta and s = \sin\theta, u_{nn} is of order 2^n times larger than the smallest singular value for small \theta and U_n(\theta) is invariant under QR factorization with column pivoting.

In practice, column pivoting reduces the efficiency of Householder QR factorization because it limits the amount of the computation that can be expressed in terms of matrix multiplication. This has motivated the development of methods that select the pivot columns using randomized projections. These methods gain speed and produce factorizations of similar rank-revealing quality, though they give up the inequalities (\dagger).

It is known that at least one permutation matrix \Pi exists such that the QR factorization of A\Pi is rank-revealing. Computing such a permutation is impractical, but heuristic algorithms for producing an approximate rank-revealing factorization are available.

References

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What is the Cayley–Hamilton Theorem?

The Cayley–Hamilton Theorem says that a square matrix A satisfies its characteristic equation, that is p(A) = 0 where p(t) = \det(tI-A) is the characteristic polynomial. This statement is not simply the substitution “p(A) = \det(A - A) = 0”, which is not valid since t must remain a scalar inside the \det term. Rather, for an n\times n A, the characteristic polynomial has the form

\notag    p(t) = t^n + a_{n-1}t^{n-1} + \cdots + a_1 t + a_0

and the Cayley–Hamilton theorem says that

\notag    p(A) = A^n + a_{n-1}A^{n-1} + \cdots + a_1 A + a_0 I = 0.

Various proofs of the theorem are available, of which we give two. The first is the most natural for anyone familiar with the Jordan canonical form. The second is more elementary but less obvious.

First proof.

Consider a 4\times 4 Jordan block with eigenvalue \lambda:

\notag     J =      \begin{bmatrix}    \lambda & 1       & 0        & 0       \\            & \lambda & 1        & 0\\            &         & \lambda  & 1\\            &         &          & \lambda         \end{bmatrix}.

We have

\notag     J - \lambda I =      \begin{bmatrix}          0 & 1       & 0        & 0       \\            &  0      & 1        & 0\\            &         & 0        & 1\\            &         &          & 0         \end{bmatrix}, \quad     (J - \lambda I)^2 =      \begin{bmatrix}            0  & 0  & 1 & 0 \\               & 0  & 0 & 1 \\               &    & 0 & 0 \\               &    &   & 0         \end{bmatrix}, \quad     (J - \lambda I)^3 =      \begin{bmatrix}            0  & 0  & 0 & 1 \\               & 0  & 0 & 0 \\               &    & 0 & 0 \\               &    &   & 0         \end{bmatrix},

and then (J - \lambda I)^4 = 0. In general, for an n \times n Jordan block J with eigenvalue \lambda, (J - \lambda I)^k is zero apart from a kth superdiagonal of ones for k\le n-1, and (J - \lambda I)^n = 0.

Let A have the Jordan canonical form A = Z JZ^{-1}, where J = \mathrm{diag}(J_1, \dots, J_k) and each J_i is an m_i\times m_i Jordan block with eigenvalue \lambda_i. The characteristic polynomial of A can be factorized as p(t) = (t-\lambda_1)^{m_1}(t-\lambda_2)^{m_2}\dots(t-\lambda_k)^{m_k}. Note that A^i = Z J^i Z^{-1} for all i, and hence q(A) = Z q(J) Z^{-1} for any polynomial q. Then

\notag  Z^{-1}p(A)Z = p(J)        = \mathrm{diag}\bigl( p(J_1), p(J_2), \dots, p(J_k) \bigr),

and p(J_i) is zero because it contains a factor (J_i - \lambda_i I\bigr)^{m_i} and this factor is zero, as noted above. Hence Z^{-1}p(A)Z = 0 and therefore p(A) = 0.

Second Proof

Recall that the adjugate \mathrm{adj}(B) of an n\times n matrix B is the transposed matrix of cofactors, where a cofactor is a signed sum of products of n-1 entries of B, and that B\mathrm{adj}(B) = \det(B)I. With B = tI - A, each entry of \mathrm{adj}(B) is a polynomial of degree n-1 in t, so B^{-1} = \mathrm{adj}(B)/\det(B) can be written

\notag    (tI - A)^{-1} = \displaystyle\frac{\mathrm{adj}(tI - A)}{\det(tI - A)}                  = \displaystyle\frac{t^{n-1}C_{n-1} + \cdots + tC_1 + C_0}{p(t)},

for some matrices C_{n-1}, \dots, C_0 not depending on t. Rearranging, we obtain

\notag   (tI - A) \bigl(t^{n-1}C_{n-1} + \cdots + tC_1 + C_0 \bigr) =  p(t)I,

and equating coefficients of t^n, …, t^0 gives

\notag \begin{aligned}   C_{n-1} &= I,\\   C_{n-2} - A C_{n-1} &= a_{n-1}I,\\          & \vdots\\   C_0 - A C_1 &= a_1 I,\\   - A C_0 &= a_0I. \end{aligned}

Premultiplying the first equation by A^n, the second by A^{n-1}, and so on, and adding, gives

\notag     0 = A^n + a_{n-1}A^{n-1} + \cdots + a_1 A + a_0 I = p(A),

as required. This proof is by Buchheim (1884).

Applications and Generalizations

A common use of the Cayley–Hamilton theorem is to show that A^{-1} is expressible as a linear combination of I, A, …, A^{n-1}. Indeed for a nonsingular A, p(A) = 0 implies that

\notag    A^{-1} = -\displaystyle\frac{1}{a_0}             \bigl( A^{n-1} + a_{n-1}A^{n-2} + \cdots + a_1 I \bigr),

since a_0 = \det(A) \ne 0.

Similarly, A^k for any k \ge n can be expressed as a linear combination of I, A, …, A^{n-1}. An interesting implication is that any matrix power series is actually a polynomial in the matrix. Thus the matrix exponential \mathrm{e}^A = I + A + A^2/2! + \cdots can be written \mathrm{e}^A = c_{n-1} A^{n-1} + \cdots + C_1A + c_0I for some scalars c_{n-1}, …, c_0. However, the c_i depend on A, which reduces the usefulness of the polynomial representation. A rare example of an explicit expression of this form is Rodrigues’s formula for the exponential of a skew-symmetric matrix A \in \mathbb{R}^{3\times 3}:

\notag      \mathrm{e}^A = I + \displaystyle\frac{\sin\theta}{\theta} A +                \displaystyle\frac{1-\cos\theta}{\theta^2} A^2,

where \theta = \sqrt{\|A\|_F^2/2}.

Cayley used the Cayley–Hamilton theorem to find square roots of a 2\times 2 matrix. If X^2 = A then applying the theorem to X gives X^2 - \mathrm{trace}(X)X + \det(X)I = 0, or

\notag \qquad\qquad\qquad\qquad   A - \mathrm{trace}(X)X + \det(X)I = 0, \qquad\qquad\qquad\qquad (*)

which gives

X = \displaystyle\frac{A + \det(X)I}{\mathrm{trace}(X)}.

Now \det(X) = \sqrt{\det(A)} and taking the trace in (*) gives an equation for \mathrm{trace}(X), leading to

\notag         X = \displaystyle\frac{ A + \sqrt{\det(A)} \,I}                  {\sqrt{\mathrm{trace}(A) + 2 \sqrt{\det(A)}}}.

With appropriate choices of signs for the square roots this formula gives all four square roots of A when A has distinct eigenvalues, but otherwise the formula can break down.

Expressions obtained from the Cayley–Hamilton theorem are of little practical use for general matrices, because algorithms that compute the coefficients a_i of the characteristic polynomial are typically numerically unstable.

The Cayley–Hamilton theorem has been generalized in various directions. The theorem can be interpreted as saying that the powers A^i for all nonnegative i generate a vector space of dimension at most n. Gerstenhaber (1961) proved that if A and B are two commuting n\times n matrices then the matrices A^iB^j, for all nonnegative i and j, generate a vector space of dimension at most n. It is conjectured that this result extends to three matrices.

Historical Note

The Cayley–Hamilton theorem appears in the 1858 memoir in which Cayley introduced matrix algebra. Cayley gave a proof for n = 2 and stated that he had verified the result for n = 3, adding “I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree.” Hamilton had proved the result for quaternions in 1853. Cayley actually discovered a more general version of the Cayley–Hamilton theorem, which appears in an 1857 letter to Sylvester but not in any of his published work: if the square matrices A and B commute and f(x,y) = \det(x A - y B) then f(B,A) = 0.

References

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What Is the CS Decomposition?

The CS (cosine-sine) decomposition reveals close relationships between the singular value decompositions (SVDs) of the blocks an orthogonal matrix expressed in block 2\times 2 form. In full generality, it applies when the diagonal blocks are not necessarily square. We focus here mainly on the most practically important case of square diagonal blocks.

Let Q\in\mathbb{R}^{n \times n} be orthogonal and suppose that n = 2p is even and Q is partitioned into four equally sized blocks:

\notag    Q =    \begin{array}[b]{@{\mskip35mu}c@{\mskip-20mu}c@{\mskip-10mu}c@{}}    \scriptstyle p &    \scriptstyle p &    \\    \multicolumn{2}{c}{        \left[\begin{array}{c@{~}c@{~}}                  Q_{11}& Q_{12} \\                  Q_{21}& Q_{22} \\              \end{array}\right]}    & \mskip-12mu\          \begin{array}{c}              \scriptstyle p \\              \scriptstyle p              \end{array}    \end{array}.

Then there exist orthogonal matrices U_1,U_2,V_1,V_2\in\mathbb{R}^{p \times p} such that

\notag    \begin{bmatrix}  U_1^T & 0\\                          0   & U_2^T    \end{bmatrix}    \begin{bmatrix}  Q_{11} & Q_{12}\\                          Q_{21} & Q_{22}    \end{bmatrix}    \begin{bmatrix}  V_1 & 0\\                          0   & V_2    \end{bmatrix}    =    \begin{array}[b]{@{\mskip36mu}c@{\mskip-13mu}c@{\mskip-10mu}c@{}}    \scriptstyle p &    \scriptstyle p &    \\    \multicolumn{2}{c}{        \left[\begin{array}{@{\mskip3mu}rr@{~}}                  C &    S         \\                 -S &    C              \end{array}\right]}    & \mskip-12mu\          \begin{array}{c}              \scriptstyle p \\              \scriptstyle p              \end{array}    \end{array},

where C = \mathrm{diag}(c_i) and S = \mathrm{diag}(s_i) with c_i \ge 0, s_i \ge 0, and c_i^2 + s_i^2 = 1 for all i. This CS decomposition comprises four SVDs:

\notag   \begin{alignedat}{2}   Q_{11} &= U_1CV_1^T,   &\quad Q_{12} &= U_1 S V_2^T, \\   Q_{21} &= U_2 (-S) V_1^T,   &\quad Q_{22} &= U_2C V_2^T.   \end{alignedat}

(Strictly speaking, for Q_{21} we need to move the minus sign from S to U_2 or V_1 to obtain an SVD.) The orthogonality ensures that there are only four different singular vector matrices instead of eight, and it makes the singular values of the blocks closely linked. We also obtain SVDs of four cross products of the blocks: Q_{11}^TQ_{12} = V_1^T CS V_2^T, etc.

Note that for p = 1, the CS decomposition reduces to the fact that any 2\times 2 orthogonal matrix is of the form \left[\begin{smallmatrix} c & s \\ -s & c \end{smallmatrix}\right] (a rotation ) up to multiplication of a row or column by -1.

A consequence of the decomposition is that Q_{11} and Q_{22} have the same 2-norms and Frobenius norms, as do their inverses if they are nonsingular. The same is true for Q_{12} and Q_{21}.

Now we drop the requirement that n is even and consider diagonal blocks of different sizes:

\notag    Q =    \begin{array}[b]{@{\mskip33mu}c@{\mskip-16mu}c@{\mskip-10mu}c@{}}    \scriptstyle p &    \scriptstyle n-p &    \\    \multicolumn{2}{c}{        \left[\begin{array}{c@{~}c@{~}}                  Q_{11}& Q_{12} \\                  Q_{21}& Q_{22} \\              \end{array}\right]}    & \mskip-12mu\          \begin{array}{c}              \scriptstyle p \\              \scriptstyle n-p              \end{array}    \end{array}, \quad p \le \displaystyle\frac{n}{2}.

The CS decomposition now has the form

\notag    \begin{bmatrix}  U_1^T & 0\\                          0   & U_2^T    \end{bmatrix}    \begin{bmatrix}  Q_{11} & Q_{12}\\                          Q_{21} & Q_{22}    \end{bmatrix}    \begin{bmatrix}  V_1 & 0\\                          0   & V_2    \end{bmatrix}    =    \begin{array}[b]{@{\mskip35mu}c@{\mskip30mu}c@{\mskip-10mu}c@{}c}    \scriptstyle p &    \scriptstyle p &    \scriptstyle n-2p &    \\    \multicolumn{3}{c}{    \left[\begin{array}{c@{~}|c@{~}c}    C &    S      & 0   \\    \hline   -S &    C      & 0   \\    0 &    0      & I_{n-2p}    \end{array}\right]}    & \mskip-12mu    \begin{array}{c}    \scriptstyle p \\    \scriptstyle p \\    \scriptstyle n-2p    \end{array}    \end{array},

with U_1, U_2, C, and S, and V_1 and V_2 (both now (n-p) \times )n-p)), having the same properties as before. The new feature for p < n/2 is the identity matrix in the bottom right-hand corner on the right-hand side. Here is an example with p = 2 and n=5, with elements shown to two decimal places:

\notag \begin{aligned} \left[\begin{array}{rr|rrr}  0.71  & -0.71  &  0     &  0     &  0     \\ -0.71  & -0.71  &  0     &  0     &  0     \\\hline  0     &  0     &  0.17  &  0.61  & -0.78  \\  0     &  0     & -0.58  & -0.58  & -0.58  \\  0     &  0     & -0.80  &  0.54  &  0.25  \\  \end{array}\right] \left[\begin{array}{rr|rrr} -0.60  & -0.40  & -0.40  & -0.40  & -0.40  \\  0.40  &  0.60  & -0.40  & -0.40  & -0.40  \\\hline  0.40  & -0.40  &  0.60  & -0.40  & -0.40  \\  0.40  & -0.40  & -0.40  &  0.60  & -0.40  \\  0.40  & -0.40  & -0.40  & -0.40  &  0.60  \\ \end{array}\right] \\ \times \left[\begin{array}{rr|rrr} -0.71  &  0.71  &  0     &  0     &  0     \\ -0.71  & -0.71  &  0     &  0     &  0     \\\hline  0     &  0     &  0.17  &  0.58  & -0.80  \\  0     &  0     &  0.61  &  0.58  &  0.54  \\  0     &  0     & -0.78  &  0.58  &  0.25  \\ \end{array}\right] = \left[\begin{array}{rr|rrr}  1.00  &  0     &  0     &  0     &  0     \\  0     &  0.20  &  0     &  0.98  &  0     \\\hline  0     &  0     &  1.00  &  0     &  0     \\  0     & -0.98  &  0     &  0.20  &  0     \\  0     &  0     &  0     &  0     &  1.00  \\ \end{array}\right]. \end{aligned}

We mention two interesting consequences of the CS decomposition.

  • With p=1: if q_{11} = 0 then Q_{22} is singular.
  • For unequally sized diagonal blocks it is no longer always true that Q_{11} and Q_{22} have the same norms, but their inverses do: \|Q_{11}^{-1}\|_2 = \|Q_{22}^{-1}\|_2 = 1/\min_ic_i \ge 1. When p = 1, this relation becomes \|Q_{22}^{-1}\|_2 = 1/|q_{11}|.

The CS decomposition also exists for a rectangular matrix with orthonormal columns,

\notag    Q =    \begin{array}[b]{@{\mskip-25mu}c@{\mskip-20mu}c@{}}    \scriptstyle n    \\    \multicolumn{1}{c}{        \left[\begin{array}{@{}c@{}}                  Q_{1}\\                  Q_{2}              \end{array}\right]}    & \mskip-12mu\          \begin{array}{c}              \scriptstyle p \\              \scriptstyle q          \end{array}    \end{array}, \quad p\ge n, \quad q \ge n.

Now the decomposition takes the form

\notag    \begin{bmatrix}  U_1^T & 0\\                          0   & U_2^T    \end{bmatrix}    \begin{bmatrix}  Q_{1}\\                     Q_{2}    \end{bmatrix}    V    =    \begin{array}[b]{@{\mskip-25mu}c@{\mskip-20mu}c@{}}    \scriptstyle n    \\    \multicolumn{1}{c}{        \left[\begin{array}{c@{~}}                  C\\                  S              \end{array}\right]}    & \mskip-12mu\          \begin{array}{c}              \scriptstyle p \\              \scriptstyle q          \end{array}    \end{array},

where U_1\in\mathbb{R}^{p\times p}, U_2\in\mathbb{R}^{q\times q}, and V\in\mathbb{R}^{n\times n} are orthogonal and C and S have the same form as before except that they are rectangular.

The most general form of the CS decomposition is for an orthogonal matrix with diagonal blocks that are not square. Now the matrix on the right-hand side has a more complicated block structure (see the references for details).

The CS decomposition arises in measuring angles and distances between subspaces. These are defined in terms of the orthogonal projectors onto the subspaces, so singular values of orthonormal matrices naturally arise.

Software for computing the CS decomposition is available in LAPACK, based on an algorithm of Sutton (2009). We used a MATLAB interface to it, available on MathWorks File Exchange, for the numerical example. Note that the output of this code is not quite in the form in which we have presented the decomposition, so some post-processing is required to achieve it.

References

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What Is the Singular Value Decomposition?

A singular value decomposition (SVD) of a matrix A\in\mathbb{R}^{m\times n} is a factorization

\notag     A = U\Sigma V^T,

where U\in\mathbb{R}^{m\times m} and V\in\mathbb{R}^{n\times n} are orthogonal, \Sigma = \mathrm{diag}(\sigma_1,\dots, \sigma_p)\in\mathbb{R}^{m\times n}, where p = \min(m,n), and \sigma_1\ge \sigma_2\ge \cdots \ge \sigma_p \ge 0.

Partition U =[ u_1,\dots,u_m] and V = [v_1,\dots, v_n]. The \sigma_i are called the singular values of A and the u_i and v_i are the left and right singular vectors. We have Av_i = \sigma_i u_i, i = 1 \colon p. The matrix \Sigma is unique but U and V are not. The form of \Sigma is

\notag  \Sigma =       \left[\begin{array}{ccc}\sigma_1&&\\ &\ddots&\\& &\sigma_n\\\hline         &\rule{0cm}{15pt} \text{\Large 0} &      \end{array}\right]      \mathrm{for}~ m \ge n, \quad    \Sigma =     \begin{bmatrix}         \begin{array}{ccc|c@{\mskip5mu}}\sigma_1&&\\ &\ddots&               & \text{\Large 0}           \\& &\sigma_m\end{array}\\      \end{bmatrix} \mathrm{for}~ m \le n

Here is an example, in which the entries of A have been specially chosen to give simple forms for the elements of the factors:

\notag A = \left[\begin{array}{rr} 0 & \frac{4}{3}\\[\smallskipamount]                        -1 & -\frac{5}{3}\\[\smallskipamount]                        -2 & -\frac{2}{3} \end{array}\right] = \underbrace{ \displaystyle\frac{1}{3} \left[\begin{array}{rrr} 1 & -2 & -2\\ -2 & 1 & -2\\ -2 & -2 & 1 \end{array}\right] }_U \mskip5mu \underbrace{ \left[\begin{array}{cc} 2\,\sqrt{2} & 0\\ 0 & \sqrt{2}\\ 0 & 0 \end{array}\right] }_{\Sigma} \mskip5mu \underbrace{ \displaystyle\frac{1}{\sqrt{2}} \left[\begin{array}{cc} 1 & 1\\ 1 & -1  \end{array}\right] }_{V^T}.

The power of the SVD is that it reveals a great deal of useful information about norms, rank, and subspaces of a matrix and it enables many problems to be reduced to a trivial form.

Since U and V are nonsingular, \mathrm{rank}(A) = \mathrm{rank}(\Sigma) = r, where r \le p is the number of nonzero singular values. Since the 2-norm and Frobenius norm are invariant under orthogonal transformations, \|A\| = \|\Sigma\| for both norms, giving

\notag   \|A\|_2 = \sigma_1, \quad   \|A\|_F = \Bigl(\displaystyle\sum_{i=1}^r \sigma_i^2\Bigr)^{1/2},

and hence \|A\|_2 \le \|A\|_F \le r^{1/2} \|A\|_2. The range space and null space of A are given in terms of the columns of U and V by

\notag \begin{aligned} \mathrm{null}(A)  &= \mathrm{span} \{ v_{r+1}, \dots,v_n \},\\ \mathrm{range}(A) &= \mathrm{span} \{u_1,u_2,\dots, u_r\}. \end{aligned}

We can write the SVD as

\notag \qquad\qquad     A = \begin{bmatrix} u_1, u_2 \dots, u_r \end{bmatrix}         \mathrm{diag}(\sigma_1,\dots, \sigma_r)        \begin{bmatrix} v_1^T\\ v_2^T\\ \vdots\\ v_r^T \end{bmatrix}     = \displaystyle\sum_{i=1}^{r} \sigma_i u_i v_i^T,  \qquad\qquad(*)

which expresses A as a sum of r rank-1 matrices, the ith of which has 2-norm \sigma_i. The famous Eckart–Young theorem (1936) says that

\notag  \min_{\mathrm{rank}(B) = k} \|A-B\|_q =  \begin{cases}      \sigma_{k+1},                                & q = 2,  \\      \Bigl(\sum_{i=k+1}^r \sigma_i^2\Bigr)^{1/2}, & q = F,   \end{cases}

and that the minimum is attained at

\notag    A_k = U D_k V^T, \quad    D_k = \mathrm{diag}(\sigma_1, \dots, \sigma_k, 0, \dots, 0).

In other words, truncating the sum (*) after k < r terms gives the best rank-k approximation to A in both the 2-norm and the Frobenius norm. In particular, this result implies that when A has full rank the distance from A to the nearest rank-deficient matrix is \sigma_r.

Relations with Symmetric Eigenvalue Problem

The SVD is not directly related to the eigenvalues and eigenvectors of A. However, for m\ge n, A = U \Sigma V^T implies

\notag  A^T\!A = V \mathrm{diag}(\sigma_1^2,\dots,\sigma_n^2) V^T, \quad  AA^T = U \mathrm{diag}(\sigma_1^2,\dots,\sigma_n^2,\underbrace{0,\dots,0}_{m-n}) U^T,

so the singular values of A are the square roots of the eigenvalues of the symmetric positive semidefinite matrices A^T\!A and AA^T (modulo m-n zeros in the latter case), and the singular vectors are eigenvectors. Moreover, the eigenvalues of the (m+n)\times (m+n) matrix

\notag     C = \begin{bmatrix}               0 & A \\               A^T & 0         \end{bmatrix}

are plus and minus the singular values of A, together with |m-n| additional zeros if m \ne n, and the eigenvectors of C and the singular vectors of A are also related.

Consequently, by applying results or algorithms for the eigensystem of a symmetric matrix to A^T\!A, AA^T, or C one obtains results or algorithms for the singular value decomposition of A.

Connections with Other Problems

The pseudoinverse of a matrix A\in\mathbb{R}^{n\times n} can be expressed in terms of the SVD as

\notag    A^+ = V\mathrm{diag}(\sigma_1^{-1},\dots,\sigma_r^{-1},0,\dots,0)U^T.

The least squares problem \min_x \|b - Ax\|_2, where A\in\mathbb{R}^{m\times n} with m \ge n is solved by x = A^+b, and when A is rank-deficient this is the solution of minimum 2-norm. For m < n this is an underdetermined system and x = A^+b gives the minimum 2-norm solution.

We can write A = U\Sigma V^T = UV^T \cdot V \Sigma V^T \equiv PQ, where P is orthogonal and Q is symmetric positive semidefinite. This decomposition A = PQ is the polar decomposition and Q = (A^T\!A)^{1/2} is unique. This connection between the SVD and the polar decomposition is useful both theoretically and computationally.

Applications

The SVD is used in a very wide variety of applications—too many and varied to attempt to summarize here. We just mention two.

The SVD can be used to help identify to which letters vowels and consonants have been mapped in a substitution cipher (Moler and Morrison, 1983).

An inverse use of the SVD is to construct test matrices by forming a diagonal matrix of singular values from some distribution then pre- and post-multiplying by random orthogonal matrices. The result is matrices with known singular values and 2-norm condition number that are nevertheless random. Such “randsvd” matrices are widely used to test algorithms in numerical linear algebra.

History and Computation

The SVD was introduced independently by Beltrami in 1873 and Jordan in 1874. Golub popularized the SVD as an essential computational tool and developed the first reliable algorithms for computing it. The Golub–Reinsch algorithm, dating from the late 1960s and based on bidiagonalization and the QR algorithm, is the standard way to compute the SVD. Various alternatives are available; see the references.

References

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