The Big Six Matrix Factorizations

Six matrix factorizations dominate in numerical linear algebra and matrix analysis: for most purposes one of them is sufficient for the task at hand. We summarize them here.

For each factorization we give the cost in flops for the standard method of computation, stating only the highest order terms. We also state the main uses of each factorization.

For full generality we state factorizations for complex matrices. Everything translates to the real case with “Hermitian” and “unitary” replaced by “symmetric” and “orthogonal”, respectively.

The terms “factorization” and “decomposition” are synonymous and it is a matter of convention which is used. Our list comprises three factorization and three decompositions.

Recall that an upper triangular matrix is a matrix of the form

\notag   R =   \begin{bmatrix}   r_{11} & r_{12} & \dots & r_{1n}\\          & r_{22} & \dots & r_{2n}\\          &        & \ddots& \vdots\\          &        &       & r_{nn}   \end{bmatrix},

and a lower triangular matrix is the transpose of an upper triangular one.

Cholesky Factorization

Every Hermitian positive definite matrix A\in\mathbb{C}^{n\times n} has a unique Cholesky factorization A = R^*R, where R\in\mathbb{C}^{n\times n} is upper triangular with positive diagonal elements.

Cost: n^3/3 flops.

Use: solving positive definite linear systems.

LU Factorization

Any matrix A\in\mathbb{C}^{n\times n} has an LU factorization PA = LU, where P is a permutation matrix, L is unit lower triangular (lower triangular with 1s on the diagonal), and U is upper triangular. We can take P = I if the leading principal submatrices A(1\colon k, 1\colon k), k = 1\colon n-1, of A are nonsingular, but to guarantee that the factorization is numerically stable we need A to have particular properties, such as diagonal dominance. In practical computation we normally choose P using the partial pivoting strategy, which almost always ensures numerically stable.

Cost: 2n^3/3 flops.

Use: solving general linear systems.

QR Factorization

Any matrix A\in\mathbb{C}^{m\times n} with m\ge n has a QR factorization A = QR, where Q\in \mathbb{C}^{m\times m} is unitary and R is upper trapezoidal, that is, R = \left[\begin{smallmatrix} R_1 \\ 0\end{smallmatrix}\right], where R_1\in\mathbb{C}^{n\times n} is upper triangular.

Partitioning Q = [Q_1~Q_2], where Q_1\in\mathbb{C}^{m\times n} has orthonormal columns, gives A = Q_1R_1, which is the reduced, economy size, or thin QR factorization.

Cost: 2(n^2m-n^3/3) flops for Householder QR factorization. The explicit formation of Q (which is not usually necessary) requires a further 4(m^2n-mn^2+n^3/3) flops.

Use: solving least squares problems, computing an orthonormal basis for the range space of A, orthogonalization.

Schur Decomposition

Any matrix A\in\mathbb{C}^{n\times n} has a Schur decomposition A = QTQ^*, where Q is unitary and T is upper triangular. The eigenvalues of A appear on the diagonal of T. For each k, the leading k columns of Q span an invariant subspace of A.

For real matrices, a special form of this decomposition exists in which all the factors are real. An upper quasi-triangular matrix R is a block upper triangular with whose diagonal blocks R_{ii} are either 1\times1 or 2\times2. Any A\in\mathbb{R}^{n\times n} has a real Schur decomposition A = Q R Q^T, where Q is real orthogonal and R is real upper quasi-triangular with any 2\times2 diagonal blocks having complex conjugate eigenvalues.

Cost: 25n^3 flops for Q and T (or R) by the QR algorithm; 10n^3 flops for T (or R) only.

Use: computing eigenvalues and eigenvectors, computing invariant subspaces, evaluating matrix functions.

Spectral Decomposition

Every Hermitian matrix A\in\mathbb{C}^{n\times n} has a spectral decomposition A = Q\Lambda Q^*, where Q is unitary and \Lambda = \mathrm{diag}(\lambda_i). The \lambda_i are the eigenvalues of A, and they are real. The spectral decomposition is a special case of the Schur decomposition but is of interest in its own right.

Cost: 9n^3 for Q and D by the QR algorithm, or 4n^3\!/3 flops for D only.

Use: any problem involving eigenvalues of Hermitian matrices.

Singular Value Decomposition

Any matrix A\in\mathbb{C}^{m\times n} has a singular value decomposition (SVD)

\notag   A = U\Sigma V^*, \quad   \Sigma = \mathrm{diag}(\sigma_1,\sigma_2,\dots,\sigma_p)             \in \mathbb{R}^{m\times n},   \quad   p = \min(m,n),

where U\in\mathbb{C}^{m\times m} and V\in\mathbb{C}^{n\times n} are unitary and \sigma_1\ge\sigma_2\ge\cdots\ge\sigma_p\ge0. The \sigma_i are the singular values of A, and they are the nonnegative square roots of the p largest eigenvalues of A^*A. The columns of U and V are the left and right singular vectors of A, respectively. The rank of A is equal to the number of nonzero singular values. If A is real, U and V can be taken to be real. The essential SVD information is contained in the compact or economy size SVD A = U\Sigma V^*, where U\in\mathbb{C}^{m\times r}, \Sigma = \mathrm{diag}(\sigma_1,\dots,\sigma_r), V\in\mathbb{C}^{n\times r}, and r = \mathrm{rank}(A).

Cost: 14mn^2+8n^3 for P(:,1\colon n), \Sigma, and Q by the Golub–Reinsch algorithm, or 6mn^2+20n^3 with a preliminary QR factorization.

Use: determining matrix rank, solving rank-deficient least squares problems, computing all kinds of subspace information.

Discussion

Pivoting can be incorporated into both Cholesky factorization and QR factorization, giving \Pi^T A \Pi = R^*R (complete pivoting) and A\Pi = QR (column pivoting), respectively, where \Pi is a permutation matrix. These pivoting strategies are useful for problems that are (nearly) rank deficient as they force R to have a zero (or small) (2,2) block.

The big six factorizations can all be computed by numerically stable algorithms. Another important factorization is that provided by the Jordan canonical form, but while it is a useful theoretical tool it cannot in general be computed in a numerically stable way.

For further details of these factorizations see the articles below.

These factorizations are precisely those discussed by Stewart (2000) in his article The Decompositional Approach to Matrix Computation, which explains the benefits of matrix factorizations in numerical linear algebra.

What Is a Schur Decomposition?

A Schur decomposition of a matrix A\in\mathbb{C}^{n\times n} is a factorization A = QTQ^*, where Q is unitary and T is upper triangular. The diagonal entries of T are the eigenvalues of A, and they can be made to appear in any order by choosing Q appropriately. The columns of Q are called Schur vectors.

A subspace \mathcal{X} of \mathbb{C}^{n\times n} is an invariant subspace of A if Ax\in\mathcal{X} for all x\in\mathcal{X}. If we partition Q and T conformably we can write

\notag   A [Q_1,~Q_2] = [Q_1,~Q_2]   \begin{bmatrix}   T_{11} & T_{12} \\       0  & T_{22} \\   \end{bmatrix},

which gives A Q_1 = Q_1 T_{11}, showing that the columns of Q_1 span an invariant subspace of A. Furthermore, Q_1^*AQ_1 = T_{11}. The first column of Q is an eigenvector of A corresponding to the eigenvalue \lambda_1 = t_{11}, but the other columns are not eigenvectors, in general. Eigenvectors can be computed by solving upper triangular systems involving T - \lambda I, where \lambda is an eigenvalue.

Write T = D+N, where D = \mathrm{diag}(\lambda_i) and N is strictly upper triangular. Taking Frobenius norms gives \|A\|_F^2 = \|D\|_F^2 + \|N\|_F^2, or

\notag        \|N\|_F^2 = \|A\|_F^2 - \displaystyle\sum_{i=1}^n |\lambda_i|^2.

Hence \|N\|_F is independent of the particular Schur decomposition and it provides a measure of the departure from normality. The matrix A is normal (that is, A^*A = AA^*) if and only if N = 0. So a normal matrix is unitarily diagonalizable: A = QDQ^*.

An important application of the Schur decomposition is to compute matrix functions. The relation f(A) = Qf(T)Q^* shows that computing f(A) reduces to computing a function of a triangular matrix. Matrix functions illustrate what Van Loan (1975) describes as “one of the most basic tenets of numerical algebra”, namely “anything that the Jordan decomposition can do, the Schur decomposition can do better!”. Indeed the Jordan canonical form is built on a possibly ill conditioned similarity transformation while the Schur decomposition employs a perfectly conditioned unitary similarity, and the full upper triangular factor of the Schur form can do most of what the Jordan form’s bidiagonal factor can do.

An upper quasi-triangular matrix is a block upper triangular matrix

\notag   R =   \begin{bmatrix}   R_{11} & R_{12} & \dots & R_{1m}\\          & R_{22} & \dots & R_{2m}\\          &        & \ddots& \vdots\\          &        &       &  R_{mm}   \end{bmatrix}

whose diagonal blocks R_{ii} are either 1\times1 or 2\times2. A real matrix A\in\mathbb{R}^{n \times n} has a real Schur decomposition A = QRQ^T in which in which all the factors are real, Q is orthogonal, and R is upper quasi-triangular with any 2\times2 diagonal blocks having complex conjugate eigenvalues. If A is normal then the 2\times 2 blocks R_{ii} have the form

R_{ii} = \left[\begin{array}{@{}rr@{\mskip2mu}} a & b \\                -b & a \end{array}\right], \quad b \ne 0,

which has eigenvalues a \pm \mathrm{i}b.

The Schur decomposition can be computed by the QR algorithm at a cost of about 25n^3 flops for Q and T, or 10n^3 flops for T only.

In MATLAB, the Schur decomposition is computed with the schur function: the command [Q,T] = schur(A) returns the real Schur form if A is real and otherwise the complex Schur form. The complex Schur form for a real matrix can be obtained with [Q,T] = schur(A,'complex'). The rsf2csf function converts the real Schur form to the complex Schur form. The= ordschur function takes a Schur decomposition and modifies it so that the eigenvalues appear in a specified order along the diagonal of T.

References

Related Blog Posts

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

What Is a Permutation Matrix?

A permutation matrix is a square matrix in which every row and every column contains a single 1 and all the other elements are zero. Such a matrix, P say, is orthogonal, that is, P^TP = PP^T = I_n, so it is nonsingular and has determinant \pm 1. The total number of n\times n permutation matrices is n!.

Premultiplying a matrix by P reorders the rows and postmultiplying by P reorders the columns. A permutation matrix P that has the desired reordering effect is constructed by doing the same operations on the identity matrix.

Examples of permutation matrices are the identity matrix I_n, the reverse identity matrix J_n, and the shift matrix K_n (also called the cyclic permutation matrix), illustrated for n = 4 by

\notag   I_4 = \begin{bmatrix}   1 & 0 & 0 & 0  \\   0 & 1 & 0 & 0  \\   0 & 0 & 1 & 0  \\   0 & 0 & 0 & 1 \end{bmatrix}, \qquad   J_4 = \begin{bmatrix}   0 & 0 & 0 & 1  \\   0 & 0 & 1 & 0  \\   0 & 1 & 0 & 0  \\   1 & 0 & 0 & 0 \end{bmatrix}, \qquad   K_4 = \begin{bmatrix}   0 & 1 & 0 & 0  \\   0 & 0 & 1 & 0  \\   0 & 0 & 0 & 1  \\   1 & 0 & 0 & 0 \end{bmatrix}.

Pre- or postmultiplying a matrix by J_n reverses the order of the rows and columns, respectively. Pre- or postmultiplying a matrix by K_n shifts the rows or columns, respectively, one place forward and moves the first one to the last position—that is, it cyclically permutes the rows or columns. Note that J_n is a symmetric Hankel matrix and K_n is a circulant matrix.

An elementary permutation matrix P differs from I_n in just two rows and columns, i and j, say. It can be written P = I_n - (e_i-e_j)(e_i-e_j)^T, where e_i is the ith column of I_n. Such a matrix is symmetric and so satisfies P^2 = I_n, and it has determinant -1. A general permutation matrix can be written as a product of elementary permutation matrices P = P_1P_2\dots P_k, where k is such that \det(P) = (-1)^k.

It is easy to show that \det(\lambda I - K_n) = \lambda^n - 1, which means that the eigenvalues of K_n are 1, w, w^2, \dots, w^{n-1}, where w = \exp(2\pi\mathrm{i}/n) is the nth root of unity. The matrix K_n has two diagonals of 1s, which move up through the matrix as it is powered: K_n^i \ne I for i< n and K_n^n = I. The following animated gif superposes MATLAB spy plots of K_{64}, K_{64}^2, …, K_{64}^{64} = I_{64}.

shift_powers.gif

The shift matrix K_n plays a fundamental role in characterizing irreducible permutation matrices. Recall that a matrix A\in\mathbb{C}^{n\times n} is irreducible if there does not exist a permutation matrix P such that

\notag         P^TAP = \begin{bmatrix} A_{11} & A_{12} \\                                    0   & A_{22}                  \end{bmatrix},

where A_{11} and A_{22} are square, nonempty submatrices.

Theorem 1. For a permutation matrix P \in \mathbb{R}^{n \times n} the following conditions are equivalent.

  • P is irreducible.
  • There exists a permutation matrix Q such that Q^{-1}PQ = K_n
  • The eigenvalues of P are 1, w, w^2, \dots, w^{n-1}.

One consequence of Theorem 1 is that for any irreducible permutation matrix P, \mathrm{rank}(P - I) = \mathrm{rank}(K_n - I) = n-1.

The next result shows that a reducible permutation matrix can be expressed in terms of irreducible permutation matrices.

Theorem 2. Every reducible permutation matrix is permutation similar to a direct sum of irreducible permutation matrices.

Another notable permutation matrix is the vec-permutation matrix, which relates A\otimes B to B\otimes A, where \otimes is the Kronecker product.

A permutation matrix is an example of a doubly stochastic matrix: a nonnegative matrix whose row and column sums are all equal to 1. A classic result characterizes doubly stochastic matrices in terms of permutation matrices.

Theorem 3 (Birkhoff). A matrix is doubly stochastic if and only if it is a convex combination of permutation matrices.

In coding, memory can be saved by representing a permutation matrix P as an integer vector p, where p_i is the column index of the 1 within the ith row of P. MATLAB functions that return permutation matrices can also return the permutation in vector form. Here is an example with the MATLAB lu function that illustrates how permuting a matrix can be done using the vector permutation representation.

>> A = gallery('frank',4), [L,U,P] = lu(A); P
A =
     4     3     2     1
     3     3     2     1
     0     2     2     1
     0     0     1     1
P =
     1     0     0     0
     0     0     1     0
     0     0     0     1
     0     1     0     0
>> P*A
ans =
     4     3     2     1
     0     2     2     1
     0     0     1     1
     3     3     2     1
>> [L,U,p] = lu(A,'vector'); p
p =
     1     3     4     2
>> A(p,:)
ans =
     4     3     2     1
     0     2     2     1
     0     0     1     1
     3     3     2     1

For more on handling permutations in MATLAB see section 24.3 of MATLAB Guide.

Notes

For proofs of Theorems 1–3 see Zhang (2011, Sec. 5.6). Theorem 3 is also proved in Horn and Johnson (2013, Thm. 8.7.2).

Permutations play a key role in the fast Fourier transform and its efficient implementation; see Van Loan (1992).

References

Related Blog Posts

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

What’s New in MATLAB R2022a?

In this post I discuss some of the new features in MATLAB R2022a, focusing on ones that relate to my particular interests. See the release notes for a detailed list of the many changes in MATLAB and its toolboxes. For my articles about new features in earlier releases, see here.

Themes

MATLAB Online now has themes, including a dark theme (which is my preference). We will have to wait for a future release for themes to be supported on desktop MATLAB.

Economy Factorizations

One can now write qr(A,'econ') instead of qr(A,0) and gsvd(A,B,'econ') instead of gsvd(A,B) for the “economy size” decompositions. This is useful as the 'econ' form is more descriptive. The svd function already supported the 'econ' argument. The economy-size QR factorization is sometimes called the thin QR factorization.

Tie Breaking in the round Function

The round function, which rounds to the nearest integer, now breaks ties by rounding away from zero by default and has several other tie-breaking options (albeit not stochastic rounding). See a sequence of four blog posts on this topic by Cleve Moler starting with this one from February 2021.

Tolerances for null and orth

The null (nullspace) and orth (orthonormal basis for the range) functions now accept a tolerance as a second argument, and any singular values less than that tolerance are treated as zero. The default tolerance is max(size(A)) * eps(norm(A)). This change brings the two functions into line with rank, which already accepted the tolerance. If you are working in double precision (the MATLAB default) and your matrix has inherent errors of order 10^{-8} (for example), you might set the tolerance to 10^{-8}, since singular values smaller than this are indistinguishable from zero.

Unit Testing Reports

The unit testing framework can now generate docx, html, and pdf reports after test execution, by using the function generatePDFReport in the latter case. This is useful for keeping a record of test results and for printing them. We use unit testing in Anymatrix and have now added an option to return the results in a variable so that the user can call one of these new functions.

Checking Arrays for Special Values

Previously, if you wanted to check whether a matrix had all finite values you would need to use a construction such as all(all(isfinite(A))) or all(isfinite(A),'all'). The new allfinite function does this in one go: allfinite(A) returns true or false according as all the elements of A are finite or not, and it works for arrays of any dimension.

Similarly, anynan and anymissing check for NaNs or missing values. A missing value is a NaN for numerical arrays, but is indicated in other ways for other data types.

Linear Algebra on Multidimensional Arrays

The new pagemldivide, pagemrdivide, and pageinv functions solve linear equations and calculate matrix inverses using pages of d-dimensional arrays, while tensorprod calculates tensor products (inner products, outer products, or a combination of the two) between two d-dimensional arrays.

Animated GIFs

The append option of the exportgraphics function now supports the GIF format, enabling one to create animated GIFs (previously only multipage PDF files were supported). The key command is exportgraphics(gca,file_name,"Append",true). There are other ways of creating animated GIFs in MATLAB, but this one is particularly easy. Here is an example M-file (based on cheb3plot in MATLAB Guide) with its output below.

%CHEB_GIF  Animated GIF of Chebyshev polynomials.
%   Based on cheb3plot in MATLAB Guide.
x = linspace(-1,1,1500)';
p = 49
Y = ones(length(x),p);

Y(:,2) = x;
for k = 3:p
  Y(:,k) = 2*x.*Y(:,k-1) - Y(:,k-2);
end

delete cheby_animated.gif
a = get(groot,'defaultAxesColorOrder'); m = length(a);

for j = 1:p-1 % length(k)
    plot(x,Y(:,j),'LineWidth',1.5,'color',a(1+mod(j-1,m),:));
    xlim([-1 1]), ylim([-1 1])  % Must freeze axes.
    title(sprintf('%2.0f', j),'FontWeight','normal')
    exportgraphics(gca,"cheby_animated.gif","Append",true)
end

cheby_animated.gif

What Is the Matrix Inverse?

The inverse of a matrix A\in\mathbb{C}^{n\times n} is a matrix X\in\mathbb{C}^{n\times n} such that AX = I, where I is the identity matrix (which has ones on the diagonal and zeros everywhere else). The inverse is written as A^{-1}. If the inverse exists then A is said to be nonsingular or invertible, and otherwise it is singular.

The inverse X of A also satisfies XA = I, as we now show. The equation AX = I says that Ax_j = e_j for j=1\colon n, where x_j is the jth column of A and e_j is the jth unit vector. Hence the n columns of A span \mathbb{C}^n, which means that the columns are linearly independent. Now A(I-XA) = A - AXA = A -A = 0, so every column of I - XA is in the null space of A. But this contradicts the linear independence of the columns of A unless I - XA = 0, that is, XA = I.

The inverse of a nonsingular matrix is unique. If AX = AW = I then premultiplying by X gives XAX = XAW, or, since XA = I, X = W.

The inverse of the inverse is the inverse: (A^{-1})^{-1} = A, which is just another way of interpreting the equations AX = XA = I.

Connections with the Determinant

Since the determinant of a product of matrices is the product of the determinants, the equation AX = I implies \det(A) \det(X) = 1, so the inverse can only exist when \det(A) \ne 0. In fact, the inverse always exists when \det(A) \ne 0.

An explicit formula for the inverse is

\notag   A^{-1} = \displaystyle\frac{\mathrm{adj}(A)}{\det(A)},   \qquad (1)

where the adjugate \mathrm{adj} is defined by

\bigl(\mathrm{adj}(A)\bigr)_{ij} = (-1)^{i+j} \det(A_{ji})

and where A_{pq} denotes the submatrix of A obtained by deleting row p and column q. A special case is the formula

\notag      \begin{bmatrix}        a & b \\ c& d      \end{bmatrix}^{-1}    = \displaystyle\frac{1}{ad-bc}      \begin{bmatrix}        d & -b \\ -c & a      \end{bmatrix}.

The equation AA^{-1} = I implies \det(A^{-1}) = 1/\det(A).

Conditions for Nonsingularity

The following result collects some equivalent conditions for a matrix to be nonsingular. We denote by \mathrm{null}(A) the null space of A (also called the kernel).

Theorem 1. For A \in \mathbb{C}^{n \times n} the following conditions are equivalent to A being nonsingular:

  • \mathrm{null}(A) = \{0\},
  • \mathrm{rank}(A) = n,
  • Ax=b has a unique solution x, for any b,
  • none of the eigenvalues of A is zero,
  • \det(A) \ne 0.

A useful formula is

\notag     (AB)^{-1} = B^{-1}A^{-1}.

Here are some facts about the inverses of n\times n matrices of special types.

  • A diagonal matrix D = \mathrm{diag}(d_i) is nonsingular if d_i\ne0 for all i, and D^{-1} = \mathrm{diag}(d_i^{-1}).
  • An upper (lower) triangular matrix T is nonsingular if its diagonal elements are nonzero, and the inverse is upper (lower) triangular with (i,i) element t_{ii}^{-1}.
  • If x,y\in\mathbb{C}^n and y^*A^{-1}x \ne -1, then A + xy^* is nonsingular and

    \notag     \bigl(A + xy^*\bigr)^{-1}       = A^{-1} - \displaystyle\frac{A^{-1}x y^* A^{-1}}{1 +  y^*A^{-1}x}.

    This is the Sherman–Morrison formula.

The Inverse as a Matrix Polynomial

The Cayley-–Hamilton theorem says that a matrix satisfies its own characteristic equation, that is, if p(t) = \det(tI - A) = t^n + c_{n-1} t^{n-1} + \cdots + c_0, then p(A) = 0. In other words, A^n + c_{n-1} A^{n-1} + \cdots + c_0I = 0, and if A is nonsingular then multiplying through by A^{-1} gives (since c_0 = p(0) = (-1)^n\det(A) \ne 0)

A^{-1} = -\displaystyle\frac{1}{c_0} (A^{n-1} + c_{n-1} A^{n-2} + \cdots + c_1 I).     \qquad (2)

This means that A^{-1} is expressible as a polynomial of degree at most n-1 in A (with coefficients that depend on A).

To Compute or Not to Compute the Inverse

The inverse is an important theoretical tool, but it is rarely necessary to compute it explicitly. If we wish to solve a linear system of equations Ax = b then computing A^{-1} and then forming x = A^{-1}b is both slower and less accurate in floating-point arithmetic than using LU factorization (Gaussian elimination) to solve the system directly. Indeed, for n = 1 one would not solve 3x = 1 by computing 3^{-1} \times 1.

For sparse matrices, computing the inverse may not even be practical, as the inverse is usually dense.

If one needs to compute the inverse, how should one do it? We will consider the cost of different methods, measured by the number of elementary arithmetic operations (addition, subtraction, division, multiplication) required. Using (1), the cost is that of computing one determinant of order n and n^2 determinants of order n-1. Since computing a k\times k determinant costs at least O(k^3) operations by standard methods, this approach costs at least O(n^5) operations, which is prohibitively expensive unless n is very small. Instead one can compute an LU factorization with pivoting and then solve the systems Ax_i = e_i for the columns x_i of A^{-1}, at a total cost of 2n^3 + O(n^2) operations.

Equation (2) does not give a good method for computing A^{-1}, because computing the coefficients c_i and evaluating a matrix polynomial are both expensive.

It is possible to exploit fast matrix multiplication methods, which compute the product of two n\times n matrices in O(n^\alpha) operations for some \alpha < 3. By using a block LU factorization recursively, one can reduce matrix inversion to matrix multiplication. If we use Strassen’s fast matrix multiplication method, which has \alpha = \log_2 7 \approx 2.807, then we can compute A^{-1} in O(n^{2.807}) operations.

Slash Notation

MATLAB uses the backslash and forward slash for “matrix division”, with the meanings A \backslash B = A^{-1}B and A / B = AB^{-1}. Note that because matrix multiplication is not commutative, A \backslash B \ne A / B, in general. We have A\backslash I = I/A = A^{-1} and I\backslash A = A/I = A. In MATLAB, the inverse can be compute with inv(A), which uses LU factorization with pivoting.

Rectangular Matrices

If A is m\times n then the equation AX = I_m requires X to be n\times m, as does XA = I_n. Rank considerations show that at most one of these equations can hold if m\ne n. For example, if A = a^* is a nonzero row vector, then AX = 1 for X = a/a^*a, but XA = aa^*/a^*a\ne I. This is an example of a generalized inverse.

An Interesting Inverse

Here is a triangular matrix with an interesting inverse. This example is adapted from the LINPACK Users’ Guide, which has the matrix, with “LINPACK” replacing “INVERSE” on the front cover and the inverse on the back cover.

\notag \left[\begin{array}{ccccccc} I & N & V & E & R & S & E\\ 0 & N & V & E & R & S & E\\ 0 & 0 & V & E & R & S & E\\ 0 & 0 & 0 & E & R & S & E\\ 0 & 0 & 0 & 0 & R & S & E\\ 0 & 0 & 0 & 0 & 0 & S & E\\ 0 & 0 & 0 & 0 & 0 & 0 & E \end{array}\right]^{-1} = \left[\begin{array}{*{7}{r@{\hspace{4pt}}}} 1/I & -1/I & 0 & 0 & 0 & 0 & 0\\ 0 & 1/N & -1/N & 0 & 0 & 0 & 0\\ 0 & 0 & 1/V & -1/V & 0 & 0 & 0\\ 0 & 0 & 0 & 1/E & -1/E & 0 & 0\\ 0 & 0 & 0 & 0 & 1/R & -1/R & 0\\ 0 & 0 & 0 & 0 & 0 & 1/S & -1/S\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/E \end{array}\right].

What Is A\A?

In a recent blog post What is A\backslash A?, Cleve Moler asked what the MATLAB operation A \backslash A returns. I will summarize what backslash does in general, for A \backslash B and then consider the case B = A.

A \backslash B is a solution, in some appropriate sense, of the equation

\notag   AX = B, \quad A \in\mathbb{C}^{m\times n}           \quad X \in\mathbb{C}^{n\times p}           \quad B \in\mathbb{C}^{m\times p}. \qquad (1)

It suffices to consider the case p = 1, because backslash treats the columns independently, and we write this as

\notag  Ax = b,  \quad A \in\mathbb{C}^{m\times n}           \quad x \in\mathbb{C}^{n}           \quad b \in\mathbb{C}^{m}.

The MATLAB backslash operator handles several cases depending on the relative sizes of the row and column dimensions of A and whether it is rank deficient.

Square Matrix: m = n

When A is square, backslash returns x = A^{-1}b, computed by LU factorization with partial pivoting (and of course without forming A^{-1}). There is no special treatment for singular matrices, so for them division by zero may occur and the output may contain NaNs (in practice, what happens will usually depend on the rounding errors). For example:

>> A = [1 0; 0 0], b = [1 0]', x = A\b
A =
     1     0
     0     0
b =
     1
     0
Warning: Matrix is singular to working precision. 

x =
     1
   NaN

Backslash take advantage of various kinds of structure in A; see MATLAB Guide (section 9.3) or doc mldivide in MATLAB.

Overdetermined System: m > n

An overdetermined system has no solutions, in general. Backslash yields a least squares (LS) solution, which is unique if A has full rank. If A is rank-deficient then there are infinitely many LS solutions, and backslash returns a basic solution: one with at most \mathrm{rank}(A) nonzeros. Such a solution is not, in general, unique.

Underdetermined System: m < n

An underdetermined system has fewer equations than unknowns, so either there is no solution of there are infinitely many. In the latter case A\backslash b produces a basic solution and in the former case a basic LS solution. Example:

>> A = [1 1 1; 1 1 0]; b = [3 2]'; x = A\b
x =
   2.0000e+00
            0
   1.0000e+00

Another basic solution is [0~2~1]^T, and the minimum 2-norm solution is [1~1~1]^T.

A\A

Now we turn to the special case A\backslash A, which in terms of equation (1) is a solution to AX = A. If A = 0 then X = I is not a basic solution, so A\backslash A \ne I; in fact, 0\backslash 0  = 0 if m\ne n and it is matrix of NaNs if m = n.

For an underdetermined system with full-rank A, A\backslash A is not necessarily the identity matrix:

>> A = [1 0 1; 0 1 0], X = A\A
A =
     1     0     1
     0     1     0
X =
     1     0     1
     0     1     0
     0     0     0

But for an overdetermined system with full-rank A, A\backslash A is the identity matrix:

>> A'\A'
ans =
   1.0000e+00            0
  -1.9185e-17   1.0000e+00

Minimum Frobenius Norm Solution

The MATLAB definition of A\backslash b is a pragmatic one, as it computes a solution or LS solution to Ax = b in the most efficient way, using LU factorization (m = n) or QR factorization (m\ne n). Often, one wants the solution of minimum 2-norm, which can be expressed as A^+b, where A^+ is the pseudoinverse of A. In MATLAB, A^+b can be computed by lsqminnorm(A,b) or pinv(A)*b, the former expression being preferred as it avoids the unnecessary computation of A^+ and it uses a complete orthogonal factorization instead of an SVD.

When the right-hand side is a matrix, B, lsqminnorm(A,B) and pinv(A)*B give the solution of minimal Frobenius norm, which we write as A \backslash\backslash B. Then A\backslash\backslash A = A^+A, which is the orthogonal projector onto \mathrm{range}(A^*), and it is equal to the identity matrix when m\ge n and A has full rank. For the matrix above:

>> A = [1 0 1; 0 1 0], X = lsqminnorm(A,A)
A =
     1     0     1
     0     1     0
X =
   5.0000e-01            0   5.0000e-01
            0   1.0000e+00            0
   5.0000e-01            0   5.0000e-01

Related Blog Posts

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

What Is the Jordan Canonical Form?

How close can similarity transformations take a matrix towards diagonal form? The answer is given by the Jordan canonical form, which achieves the largest possible number of off-diagonal zero entries (Brualdi, Pei, and Zhan, 2008).

Theorem (Jordan canonical form). Any matrix A\in\mathbb{C}^{n\times n} can be expressed as

\notag \begin{aligned}    A &= ZJZ^{-1}, \quad   J = \mathrm{diag}(J_1, J_2, \dots, J_p), \\   J_k &= J_k(\lambda_k) =       \begin{bmatrix} \lambda_k & 1         &          &           \\                           & \lambda_k & \ddots   &           \\                           &           & \ddots   &    1      \\                           &           &          & \lambda_k \end{bmatrix}     \in \mathbb{C}^{m_k\times m_k}, \label{Jk} \end{aligned}

where Z is nonsingular and m_1 + m_2 + \cdots + m_p = n. The matrix J is unique up to the ordering of the blocks J_k.

The matrix J is (up to reordering of the diagonal blocks) the Jordan canonical form of A (or the Jordan form, for short).

The bidiagonal matrices J_k are called Jordan blocks. Clearly, the eigenvalues of J_k are \lambda_k repeated m_k times and J_k has a single eigenvector, e_1\in\mathbb{R}^{m_k}. Two different Jordan blocks can have the same eigenvalues.

In total, J has p linearly independent eigenvectors, and the same is true of A.

The Jordan canonical form is an invaluable tool in matrix analysis, as it provides a concrete way to prove and understand many results. However, the Jordan form can not be reliably computed in finite precision arithmetic, so it is of little use computationally, except in special cases such as when A is Hermitian or normal.

For a Jordan block J_k = J_k(\lambda_k)\in\mathbb{C}^{m_k\times m_k} we have

\notag \begin{aligned} J_k - \lambda_k I    &=  \begin{bmatrix} 0         & 1         &          &           \\                           & 0         & \ddots   &           \\                           &           & \ddots   &    1      \\                           &           &          & 0         \end{bmatrix}, \quad (J_k - \lambda_k I)^2    =  \begin{bmatrix} 0         & 0         & 1        &        &           \\                           & 0         & 0        &    \ddots   &      \\                           &           & \ddots   &  \ddots& 1   \\                           &           &          & \ddots & 0     \\                           &           &          &      & 0    \end{bmatrix},\\ \dots,\quad (J_k - \lambda_k I)^{m_k-1}    &=  \begin{bmatrix} 0         & 0         & \dots    &    1      \\                           & 0         & \dots    &    0      \\                           &           & \ddots   &    \vdots \\                           &           &          & 0          \end{bmatrix}, \quad   (J_k - \lambda_k I)^{m_k} = 0. \qquad (*)\notag \end{aligned}

The superdiagonal of ones moves up to the right with each increase in the index of the power, until it disappears off the top corner of the matrix.

It is easy to see that (A - \lambda I_n)^{j} = Z(J - \lambda I_n)^{j} Z^{-1}      = Z\mathrm{diag}\bigl((J_k(\lambda_k) - \lambda I_{m_k})^{j}\bigr) Z^{-1}, and so

\mathrm{rank}( (A - \lambda I_n)^{j} ) = \sum_{k = 1}^p\mathrm{rank}\bigl( (J_k(\lambda_k) - \lambda I_{m_k})^{j} \bigr).

For \lambda = \lambda_k, these quantities provide information about the size of the Jordan blocks associated with \lambda_k. To be specific, let

\notag    d_j = \mathrm{rank}( (A - \lambda_kI_n)^{j}), \quad j\ge 1, \quad \quad d_0 = n

and

\notag   \omega_j = d_{j-1} - d_j, \quad j \ge 1.

By considering the equations (*) above, it can be shown that \omega_j is the number of Jordan blocks of size at least j in which \lambda_k appears. Moreover, the number of Jordan blocks of size j is \omega_j - \omega_{j+1} = d_{j-1} - 2d_j + d_{j+1}. Therefore if we know the eigenvalues and the ranks of (A - \lambda_k I_n)^j for each eigenvalue \lambda_k and appropriate j then we can determine the Jordan structure. As an important special case, if \mathrm{rank}(A - \lambda_k I_n) = n-1 then we know that \lambda_k appears in a single Jordan block. The sequence of \omega_j is known as the Weyr characteristic, and it satisfies \omega_1 \ge \omega_2 \ge \cdots.

As an example of a matrix for which we can easily deduce the Jordan form consider the nilpotent matrix B = \bigl[\begin{smallmatrix} 0_r & I_r\\0_r & 0_r \end{smallmatrix}\bigr], for which B^2 = 0 and all the eigenvalues are zero. Since \mathrm{rank}(B) = r, we have d_0 = 2r, d_1 = r, and d_2 = 0. Hence \omega_1 = 0 and \omega_2 = r, so there are r 2\times 2 Jordan blocks. (In fact, A can be permuted into Jordan form by a similarity transformation.)

Here is an example with A the 11\times 11 matrix anymatrix('core/collatz',11).

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

We have \mathrm{rank}(A) = 10 and \mathrm{rank}(A-2I) = 10, so 0 and 2 are simple eigenvalues. All the other eigenvalues are 1 and they have the following d_j and \omega_j values:

\notag \begin{array}{cccc} j &  d_j &\omega_j &\omega_j - \omega_{j+1}\\\hline  0 & 11 &     &     \\  1 &  7 &   4 &   2 \\  2 &  5 &   2 &   1 \\  3 &  4 &   1 &   0 \\  4 &  3 &   1 &   0 \\  5 &  2 &   1 &   1 \\  6 &  2 &   0 &   0 \\ \end{array}

We conclude that the eigenvalue 1 occurs in one block of order 5, one block of order 2, and two blocks of order 1.

A matrix and its transpose have the same Jordan form. One way to see this is to note that A = ZJZ^{-1} implies

A^T = Z^{-T}J^TZ^T      = Z^{-T}P \cdot PJ^TP \cdot PZ^T      = (ZP)^{-T}J \,(ZP)^T,

where P is the identity matrix with the its columns reversed. A consequence is that A and A^T are similar.

Real Jordan Form

A version of the Jordan form with Z and J real exists for A\in\mathbb{R}^{n\times n}. The main change is how complex eigenvalues are represented. Since the eigenvalues now occur in complex conjugate pairs \lambda and \overline{\lambda}, and each of the pair has the same Jordan structure (which follows from the fact that a matrix and its complex conjugate have the same rank), pairs of Jordan blocks corresponding to \lambda and \overline{\lambda} are combined into a real block of twice the size. For example, Jordan blocks

\notag  \begin{bmatrix} \lambda & 1 \\ 0 & \lambda \end{bmatrix}, \;  \begin{bmatrix}\,\overline{\lambda} & 1 \\ 0 & \overline{\lambda} \end{bmatrix}  \in\mathbb{C}^{2 \times 2}

become

\notag     \left[\begin{array}{@{\mkern3mu}rr|rr@{\mkern7mu}}          a & b & 1 & 0 \\         -b & a & 0 & 1 \\\hline          0 & 0 & a & b \\          0 & 0 &-b & a          \end{array}\right]  \in\mathbb{R}^{4 \times 4}

in the real Jordan form, where \lambda = a + \mathrm{i} b. Note that the eigenvalues of \bigl[\begin{smallmatrix} a & b \\ -b & a \end{smallmatrix}\bigr] are a \pm \mathrm{i} b.

Notes

Proofs of the Jordan canonical form and its real variant can be found in many textbooks. See also Brualdi (1987) and Fletcher and Sorensen (1983), who give proofs that go via the Schur decomposition.

References

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

What Is the Second Difference Matrix?

The second difference matrix is the tridiagonal matrix T_n with diagonal elements 2 and sub- and superdiagonal elements -1:

\notag    T_n = \left[    \begin{array}{@{}*{4}{r@{\mskip10mu}}r}                 2  & -1 &        &        &    \\                 -1 & 2  & -1     &        &    \\[-5pt]                    & -1 & 2      & \ddots &    \\                    &    & \ddots & \ddots & -1 \\                    &    &        & -1     & 2    \end{array}\right] \in\mathbb{R}^{n\times n}.

It arises when a second derivative is approximated by the central second difference f''(x) \approx (f(x+h) -2 f(x) + f(x-h))/h^2. (Accordingly, the second difference matrix is sometimes defined as -T_n.) In MATLAB, T_n can be generated by gallery('tridiag',n), which is returned as a aparse matrix.

This is Gil Strang’s favorite matrix. The photo, from his home page, shows a birthday cake representation of the matrix.

strang_second_diff_cake.jpg

The second difference matrix is symmetric positive definite. The easiest way to see this is to define the full rank rectangular matrix

\notag  L_n = \begin{bmatrix}                1  &      &        &  \\                -1 &  1   &        &  \\                   & -1   & \ddots &  \\                   &      & \ddots & 1 \\                   &      &        &  -1     \end{bmatrix} \in\mathbb{R}^{(n+1)\times n}

and note that T_n = L_n^T L_n. The factorization corresponds to representing a central difference as the product of a forward difference and a backward difference. Other properties of the second difference matrix are that it is diagonally dominant, a Toeplitz matrix, and an M-matrix.

Cholesky Factorization

In an LU factorization A = LU the pivots u_{ii} are 2, 3/2, 4/3, …, (n+1)/n. Hence the pivots form a decreasing sequence tending to 1 as n\to\infty. The diagonal of the Cholesky factor contains the square roots of the pivots. This means that in the Cholesky factorization A = R^*R with R upper bidiagonal, r_{nn} \to 1 and r_{n,n-1}\to -1 as n\to\infty.

Determinant, Inverse, Condition Number

Since the determinant is the product of the pivots, \det(T_n) = n+1.

The inverse of T_n is full, with (i,j) element i(n-j+1)/(n+1) for j\ge i. For example,

\notag   T_5^{-1} = \displaystyle\frac{1}{6}  \begin{bmatrix} 5 & 4 & 3 & 2 & 1\\ 4 & 8 & 6 & 4 & 2\\ 3 & 6 & 9 & 6 & 3\\ 2 & 4 & 6 & 8 & 4\\ 1 & 2 & 3 & 4 & 5  \end{bmatrix}.

The 2-norm condition number satisfies \kappa_2(T_n) \sim 4n^2/\pi^2 (as follows from the formula (1) below for the eigenvalues).

Eigenvalues and Eigenvectors

The eigenvalues of T_n are

\notag      \mu_k       = 4 \sin^2\Bigl(\displaystyle\frac{k \phi}{2} \Bigr),         \quad k = 1:n, \qquad (1)

where \phi = \pi/(n+1), with corresponding eigenvector

\notag   v_k = \begin{bmatrix} \sin(k\phi), &\sin(2k\phi), &\dots, &\sin(nk\phi)   \end{bmatrix}^T.

The matrix Q with

\notag    q_{ij} = \Bigl(\displaystyle\frac{2}{n+1}\Bigr)^{1/2} \sin\Bigl( \frac{ij\pi}{n+1} \Bigr)

is therefore an eigenvector matrix for T_n: Q^*AQ = \mathrm{diag}(\mu_k).

Variations

Various modifications of the second difference matrix arise and similar results can be derived. For example, consider the matrix obtained by changing the (n,n) element to 1:

\notag    \widetilde{T}_n = \left[    \begin{array}{@{}*{4}{r@{\mskip10mu}}r}                 2  & -1 &        &        &    \\                 -1 & 2  & -1     &        &    \\[-5pt]                    & -1 & 2      & \ddots &    \\                    &    & \ddots & \ddots & -1 \\                    &    &        & -1     & 1    \end{array}\right] \in\mathbb{R}^{n\times n}.

It can be shown that \widetilde{T}_n^{-1} has (i,j) element \min(i,j) and eigenvalues 4\cos( j \pi/(2n+1))^2, j=1:n.

If we perturb the (1,1) of \widetilde{T}_n to 1, we obtain a singular matrix, but suppose we perturb the (1,1) element to 3:

\notag    \widehat{T}_n = \left[    \begin{array}{@{}*{4}{r@{\mskip10mu}}r}                 3  & -1 &        &        &    \\                 -1 & 2  & -1     &        &    \\[-5pt]                    & -1 & 2      & \ddots &    \\                    &    & \ddots & \ddots & -1 \\                    &    &        & -1     & 1    \end{array}\right] \in\mathbb{R}^{n\times n}.

The inverse is \widehat{T}_n^{-1} = G/2, where G with (i,j) element 2\min(i,j)-1 is a matrix of Givens, and the eigenvalues are 4\cos((2j-1)\pi/(4n))^2, j=1:n.

Notes

The factorization T_n = L_n^TL_n is noted by Strang (2012).

For a derivation of the eigenvalues and eigenvectors see Todd (1977, p. 155 ff.). For the eigenvalues of \widetilde{T}_n see Fortiana and Cuadras (1997). Givens’s matrix is described by Newman and Todd (1958) and Todd (1977).

References

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

Related Blog Posts

What Is the Frank Matrix?

The n\times n upper Hessenberg matrix

\notag   F_n =   \left[\begin{array}{*{6}c}   n      & n-1    & n-2    & \dots  & 2      & 1 \\   n-1    & n-1    & n-2    & \dots  & 2      & 1 \\   0      & n-2    & n-2    & \dots  & 2      & 1 \\[-3pt]   \vdots & 0      & \ddots & \ddots & \vdots & 1 \\[-3pt]   \vdots & \vdots & \dots  &   2    & 2      & 1 \\   0      & 0      & \dots  &   0    & 1      & 1 \\   \end{array}\right]

was introduced by Frank in 1958 as a test matrix for eigensolvers.

Determinant

Taking the Laplace expansion about the first column, we obtain \det(F_n) = n\det(F_{n-1}) - (n-1)\det(F_{n-1})             = \det(F_{n-1}), and since \det(F_1) = 1 we have \det(F_n) = 1.

In MATLAB, the computed determinant of the matrix and its transpose can be far from the exact values of 1:

>> n = 20; F = anymatrix('gallery/frank',n); 
>> [det(F), det(F')]
ans =
   1.0000e+00  -1.4320e+01
>> n = 49; F = anymatrix('gallery/frank',n); det(F)
ans =
    -1.406934439401568e+45

This behavior illustrates the sensitivity of the determinant rather than numerical instability in the evaluation and it is very dependent on the arithmetic (different results may be obtained in different releases of MATLAB). The sensitivity can be seen by noting that

\notag     \det\bigl(F_n + \epsilon e_1e_n^T\bigr) = 1 + (-1)^{n-1} (n-1)!\epsilon,     \qquad (1)

which means that changing the (1,n) element from 1 to 1+\epsilon changes the determinant by (n-1)!\epsilon.

Inverse and Condition Number

It is easily verified that

\notag       F_n =             \begin{bmatrix}            1 & -1 &        &        &   \\                          &  1 & -1     &        &   \\                          &    & 1      & \ddots &   \\                          &    &        & \ddots & -1\\                          &    &        &        & 1        \end{bmatrix}^{-1}       \begin{bmatrix}               1 &    &        &        &   \\                      n-1 &  1 &        &        &   \\                          & n-2&  1     &        &   \\                          &    &  \ddots& \ddots &       \\                          &    &        &   1    & 1    \end{bmatrix}          \equiv U^{-1}L.       \qquad (2)

Hence F_n^{-1} = L^{-1}U is lower Hessenberg with integer entries. This factorization provides another way to see that \det(F_n) = 1.

From (1) we see that F_n + \epsilon e_1e_n^T is singular for \epsilon = (-1)^n/(n-1)!, which implies that

\notag      \kappa(F_n) \ge (n-1)! \|F_n\|

for any subordinate matrix norm, showing that the condition number grows very rapidly with n. In fact, this lower bound is within a factor 3.5 of the condition number for the 1-, 2-, and \infty-norms for n\le 20.

Eigenvalues

Denote by p_n(t) = \det(F_n - tI) the characteristic polynomial of F_n. By expanding about the first column one can show that

\label{Fnpn-rec}     p_n(t) = (1-t)p_{n-1}(t) - (n-1)t p_{n-2}(t).    \qquad (3)

Using (3), one can show by induction that

\notag       p_n(t^{-1}) = (-1)^n t^{-n} p_n(t).

This means that the eigenvalues of F_n occur in reciprocal pairs, and hence that \lambda = 1 is an eigenvalue when n is odd. It also follows that p_n is palindromic when n is even and anti-palindromic when n is odd. Examples:

>> charpoly( anymatrix('gallery/frank',6) )
ans =
     1   -21   120  -215   120   -21     1
>> charpoly( anymatrix('gallery/frank',7) )
ans =
     1   -28   231  -665   665  -231    28    -1

Since F_n has nonzero subdiagonal entries, \mathrm{rank}(F_n - t I) \ge n-1 for any t, and hence F_n is nonderogatory, that is, no eigenvalue appears in more than one Jordan block in the Jordan canonical form. It can be shown that the eigenvalues are real and positive and that they can be expressed in terms of the zeros of Hermite polynomials. Furthermore, the eigenvalues are distinct.

If each eigenvalue of an n\times n matrix undergoes a small perturbation then the determinant, being the product of the eigenvalues, undergoes a perturbation up to about n times as large. From (1) we see that a change to F_n of order \epsilon can perturb \det(F_n) by (n-1)!\epsilon, and it follows that some eigenvalues must undergo a large relative perturbation. The condition number of a simple eigenvalue is defined as the reciprocal of the cosine of the angle between its left and right eigenvectors. For the Frank matrix it is the small eigenvalues that are ill conditioned, as shown here for n = 9.

>> n = 9; F = anymatrix('gallery/frank',n);
>> [V,D,cond_evals] = condeig(F); evals = diag(D);
>> [~,k] = sort(evals,'descend');
>> [evals(k) cond_evals(k)]
ans =
   2.2320e+01   1.9916e+00
   1.2193e+01   2.3903e+00
   6.1507e+00   1.5268e+00
   2.6729e+00   3.6210e+00
   1.0000e+00   6.8615e+01
   3.7412e-01   1.5996e+03
   1.6258e-01   1.1907e+04
   8.2016e-02   2.5134e+04
   4.4803e-02   1.4762e+04

Notes

Frank found that F_n “gives our selected procedures difficulties” and that “accuracy was lost in the smaller roots”. The difficulties encountered by Frank’s codes were shown by Wilkinson to be caused by the inherent sensitivity of the eigenvalues to perturbations in the matrix.

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/index-of-what-is-articles/ and in PDF form from the GitHub repository https://github.com/higham/what-is.

What Is the Logarithmic Norm?

The logarithmic norm of a matrix A\in\mathbb{C}^{n\times n} (also called the logarithmic derivative) is defined by

\notag       \mu(A) = \displaystyle\lim_{\epsilon \to 0+} \frac{ \|I + \epsilon A\| - 1}{\epsilon},

where the norm is assumed to satisfy \|I\| = 1.

Note that the limit is taken from above. If we take the limit from below then we obtain a generally different quantity: writing \delta = -\epsilon,

\notag   \displaystyle\lim_{\epsilon \to 0-} \frac{ \|I + \epsilon A\| - 1}{\epsilon}    = \lim_{\delta \to 0+} \frac{ \|I - \delta A\| - 1}{-\delta}    = -\mu(-A).

The logarithmic norm is not a matrix norm; indeed it can be negative: \mu(-I) = -1.

The logarithmic norm can also be expressed in terms of the matrix exponential.

Lemma 1. For A\in\mathbb{C}^{n\times n},

\notag \mu(A) = \displaystyle\lim_{\epsilon\to 0+} \frac{\log\, \| \mathrm{e}^{\epsilon A}\|} {\epsilon}        = \lim_{\epsilon\to 0+} \frac{\| \mathrm{e}^{\epsilon A}\| - 1} {\epsilon}.

Properties

We give some basic properties of the logarithmic norm.

It is easy to see that

\notag     -\|A\| \le \mu(A) \le \|A\|. \qquad (1)

For z\in\mathbb{C}, we define \mathrm{sign}(z) = z/|z| for z\ne 0 and \mathrm{sign}(0) = 0.

Lemma 2. For A,B\in\mathbb{C}^{n\times n} and \lambda\in\mathbb{C},

  • \mu(\lambda A) = |\lambda| \mu\bigl(\mathrm{sign}(\lambda)A\bigr),
  • \mu(A + \lambda I) = \mu(A) + \mathrm{Re}\,\lambda,
  • \mu(A + B ) \le \mu(A) + \mu(B),
  • |\mu(A) - \mu(B)| \le \|A - B\|.

The next result shows the crucial property that \mu(A) features in an easily evaluated bound for the norm of \mathrm{e}^{At} and that, moreover, \mu(A) is the smallest constant that can appear in this bound.

Theorem 3. For A\in\mathbb{C}^{n\times n} and a consistent matrix norm,

\notag   \|\mathrm{e}^{At}\| \le \mathrm{e}^{\mu(A)t}, \quad t\ge0.   \qquad (2)

Moreover,

\notag   \mu(A) = \min\{\, \theta\in\mathbb{R}:         \|\mathrm{e}^{At}\| \le \mathrm{e}^{\theta t}               ~\mathrm{for~all}~t\ge 0 \,\}.

Proof. Given any \delta > 0, by Lemma 1 there exists h > 0 such that

\notag     \displaystyle\frac{ \| \mathrm{e}^{At}\| - 1}{t} - \mu(A) < \delta,     \quad t\in[0,h],

or

\notag     \| \mathrm{e}^{At}\| \le 1 + (\mu(A) + \delta)t         \le \mathrm{e}^{(\mu(A) + \delta)t},     \quad t\in[0,h]

(since \mathrm{e}^x \ge 1+x for all x). Then for any integer k, \|\mathrm{e}^{Atk}\| = \|( \mathrm{e}^{At})^k\| \le \| \mathrm{e}^{At}\|^k \le \mathrm{e}^{(\mu(A) + \delta)tk}, and hence \|\mathrm{e}^{At}\| \le \mathrm{e}^{(\mu(A) + \delta)t} holds for all t\in[0,\infty). Since \delta is arbitrary, it follows that \|\mathrm{e}^{At}\| \le \mathrm{e}^{\mu(A)t}.

Finally, if \| \mathrm{e}^{At}\| \le \mathrm{e}^{\theta t} for all t\ge 0 then (\| \mathrm{e}^{At}\| -1)/t\le (\mathrm{e}^{\theta t}-1)/t for all t\ge0 and taking \lim_{t\to 0+} we conclude that \mu(A) \le (d/dt) \mathrm{e}^{\theta t}\mid_{t = 0} \,= \theta.

The logarithmic norm was introduced by Dahlquist (1958) and Lozinskii (1958) in order to obtain error bounds for numerical methods for solving differential equations. The bound (2) can alternatively be proved by using differential inequalities (see Söderlind (2006)). Not only is (2) sharper than \|\mathrm{e}^{At}\| \le \mathrm{e}^{\|A\|t}, but \mathrm{e}^{\|A\|t} is increasing in t while \mathrm{e}^{\mu(A)t} potentially decays, since \mu(A) < 0 is possible.

The spectral abscissa is defined by

\notag      \alpha(A) = \max \{\, \mathrm{Re} \lambda : \lambda \in \Lambda(A)\,\},

where \Lambda(A) denotes the spectrum of A (the set of eigenvalues). Whereas the norm bounds the spectral radius (\rho(A) \le \|A\|), the logarithmic norm bounds the spectral abscissa, as shown by the next result.

Theorem 4. For A\in\mathbb{C}^{n\times n} and a consistent matrix norm,

\notag    -\mu(-A) \le \alpha(A) \le \mu(A).

Combining (1) with (2) gives

\notag   -\|A\| \le -\mu(-A) \le \alpha(A) \le \mu(A) \le \|A\|.

In view of Lemma 1, the logarithmic norm \mu(A) can be expressed as the one-sided derivative of \|\mathrm{e}^{tA}\| at t = 0, so \mu(A) determines the initial rate of change of \|\mathrm{e}^{tA}\| (as well as providing the bound \mathrm{e}^{\mu(A)t} for all t). The long-term behavior is determined by the spectral abscissa \alpha(A), since \|\mathrm{e}^{tA}\| \to 0 as t\to\infty if and only if \alpha(A) < 0, which can be proved using the Jordan canonical form.

The next result provides a bound on the norm of the inverse of a matrix in terms of the logarithmic norm.

Theorem 5. For a nonsingular matrix A\in\mathbb{C}^{n\times n} and a subordinate matrix norm, if \mu(A) < 0 or \mu(-A) < 0 then

\notag    \|A^{-1}\| \le \displaystyle\frac{1}{\max\bigl(-\mu(A),-\mu(-A)\bigr)}.     \qquad (3)

Formulas for Logarithmic Norms

Explicit formulas are available for the logarithmic norm s corresponding to the 1, 2, and \infty-norms.

Theorem 6. For A\in\mathbb{C}^{n\times n},

\notag \begin{aligned}  \mu_1(A) &= \max_j \biggl( \sum_{i\ne j} |a_{ij}| +                                   \mathrm{Re}\, a_{jj} \biggr ),\\  \mu_{\infty}(A) &= \max_i \biggl( \sum_{j\ne i} |a_{ij}| +                                   \mathrm{Re}\, a_{ii} \biggr ),\\  \mu_2(A) &= \lambda_{\max}\Bigl( \frac{A+A^*}{2} \Bigr), \qquad (4) \end{aligned}

where \lambda_{\max} denotes the largest eigenvalue of a Hermitian matrix.

Proof. We have

\notag \begin{aligned}     \mu_{\infty}(A)     &= \lim_{\epsilon \to 0+} \frac{ \|I + \epsilon A\|_{\infty} - 1}{\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ \max_i \bigl(\sum_{j\ne i} |\epsilon a_{ij}|                               + |1 + \epsilon a_{ii}| \bigr) -1}{\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ \max_i \bigl(\sum_{j\ne i} |\epsilon a_{ij}|                   + \sqrt{(1 + \epsilon\mathrm{Re}\, a_{ii})^2                   + (\epsilon\mathrm{Im}\,a_{ii})^2}\, \bigr) -1}{\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ \max_i \bigl( \sum_{j\ne i} |\epsilon a_{ij}|                   + \epsilon\mathrm{Re}\, a_{ii} + O(\epsilon^2)  \bigr)}{\epsilon}\\     &= \max_i \biggl( \sum_{j\ne i} |a_{ij}| +                                   \mathrm{Re}\, a_{ii} \biggr ). \end{aligned}

The formula for \mu_1(A) follows, since \|A\|_1 = \|A^*\|_\infty implies \mu_1(A) = \mu_{\infty}(A^*). For the 2-norm, using \|A\|_2 = \rho(A^*A)^{1/2}, we have

\notag \begin{aligned}     \mu_2(A)     &= \lim_{\epsilon \to 0+} \frac{ \|I + \epsilon A\|_2 - 1}{\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ \rho\bigl( I + \epsilon(A+A^*) +       \epsilon^2 A^*A\bigr)^{1/2} -1} {\epsilon}\\     &= \lim_{\epsilon \to 0+} \frac{ 1 + \frac{1}{2}\epsilon\lambda_{\max}(A+A^*) +                                 O(\epsilon^2)  -1} {\epsilon}\\     &= \lambda_{\max}\Bigl( \frac{A+A^*}{2} \Bigr). \end{aligned}

As a special case of (4), if A is normal, so that A = QDQ^* with Q unitary and D = \mathrm{diag}(\lambda_i), then \mu_2(A) = \max_i (\lambda_i + \overline{\lambda_i})/2 = \max_i \mathrm{Re} \lambda_i = \alpha(A).

We can make some observations about \mu_\infty(A) for A\in\mathbb{R}^{n\times n}.

  • If A\ge 0 then \mu_\infty(A) = \|A\|_\infty.
  • \mu_\infty(A) < 0 if and only if a_{ii} < 0 for all i and A is strictly row diagonally dominant.
  • For the \infty-norm the bound (3) is the same as a bound based on diagonal dominance except that it is applicable only when the diagonal is one-signed.

For a numerical example, consider the n\times n tridiagonal matrix anymatrix('gallery/lesp'), which has the form illustrated for n = 6 by

\notag     A_6 = \left[\begin{array}{cccccc}      -5 & 2 & 0 & 0 & 0 & 0\\      \frac{1}{2} & -7 & 3 & 0 & 0 & 0\\      0 & \frac{1}{3} & -9 & 4 & 0 & 0\\      0 & 0 & \frac{1}{4} & -11 & 5 & 0\\      0 & 0 & 0 & \frac{1}{5} & -13 & 6\\      0 & 0 & 0 & 0 & \frac{1}{6} & -15      \end{array}\right].

We find that \alpha(A_6) = -4.55 and \mu_2(A_6) = -4.24, and it is easy to see that \mu_1(A_n) = -4.5 and \mu_{\infty}(A_n) = -3 for all n. Therefore Theorem 4 shows that \mathrm{e}^{At}\to 0 as t\to \infty and \mu_1 gives a faster decaying bound than \mu_2 and \mu_\infty.

Now consider the subordinate matrix norm \|\cdot\|_G based on the vector norm \|x\|_G = (x^*Gx)^{1/2}, where G is a Hermitian positive definite matrix. The corresponding logarithmic norm \mu_G can be expressed as the largest eigenvalue of a Hermitian definite generalized eigenvalue problem.

Theorem 7. For A\in\mathbb{C}^{n\times n},

\notag    \mu_G(A) = \max\{\, \lambda: \det(GA + A^*G - 2\lambda G) = 0 \,\}.

Theorem 7 allows us to make a connection with the theory of ordinary differential equations (ODEs)

\notag   y' = f(t,y), \quad f: \mathbb{R} \times \mathbb{R}^n\to \mathbb{R}^n.   \qquad (5)

Let G\in\mathbb{R}^{n\times n} be symmetric positive definite and consider the inner product \langle x, y \rangle = x^*Gy and the corresponding norm defined by \|x\|_G^2 = \langle x, x \rangle = (x^*Gx)^{1/2}. It can be shown that for A\in\mathbb{R}^{n\times },

\notag   \mu_G(A) = \max_x \displaystyle\frac{\langle Ax, x\rangle}{\|x\|_G^2}.  \qquad (6)

The function f satisfies a one-sided Lipschitz condition if there is a function v(t) such that

\langle f(t,y) - f(t,z), y - z \rangle \le v(t) \|y-z\|^2

for all y,z in some region and all a\le t \le b. For the linear differential equation with f(t,y) = A(t)y in (5), using (6) we obtain

\langle f(t,y) - f(t,z), y - z \rangle =    \langle A(t)(y-z), y - z \rangle \le \mu_G(A(t)) \|y-z\|_G^2,

and so the logarithmic norm \mu_G(A(t)) can be taken as a one-sided Lipschitz constant. This observation leads to results on contractivity of ODEs; see Lambert (1991) for details.

Notes

For more on the use of the logarithmic norm in differential equations see Coppel (1965) and Söderlind (2006).

The proof of Theorem 3 is from Hinrichsen and Pritchard (2005).

References

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

  • W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations}. D. C. Heath and Company, Boston, MA. USA, 1965.
  • Germund Dahlquist. Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations. PhD thesis, Royal Inst. of Technology, Stockholm, Sweden, 1958.
  • Diederich Hinrichsen and Anthony J. Pritchard. Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness. Springer-Verlag, Berlin, Germany, 2005.
  • J. D. Lambert. Numerical Methods for Ordinary Differential Systems. The Initial Value Problem. John Wiley, Chichester, 1991.
  • Gustaf Söderlind, The Logarithmic Norm. History and Modern Theory. BIT, 46(3):631–652, 2006.
  • Torsten Ström. On Logarithmic Norms. SIAM J. Numer. Anal., 12(5):741–753, 1975.

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