What Is a Cholesky Factorization?

The Cholesky factorization of a symmetric positive definite matrix $A$ is the factorization $A = R^T\!R$ , where $R$ is upper triangular with positive diagonal elements. It is a generalization of the property that a positive real number has a unique positive square root. The Cholesky factorization always exists and the requirement that the diagonal of $R$ be positive ensures that it is unique.

As an example, the Cholesky factorization of the $4\times 4$ matrix with $(i,j)$ element $\gcd(i,j)$ (gallery('gcdmat',4) in MATLAB) is

$G_4 = \left[\begin{array}{cccc} 1 & 1 & 1 & 1\\ 1 & 2 & 1 & 2\\ 1 & 1 & 3 & 1\\ 1 & 2 & 1 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 1 & 1 & 0 & 0\\ 1 & 0 & \sqrt{2} & 0\\ 1 & 1 & 0 & \sqrt{2} \end{array}\right] \left[\begin{array}{cccc} 1 & 1 & 1 & 1\\ 0 & 1 & 0 & 1\\ 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & \sqrt{2} \end{array}\right].$

The Cholesky factorization of an $n\times n$ matrix contains $n-1$ other Cholesky factorizations within it: $A_k = R_k^TR_k$ , $k=1:n-1$ , where $A_k = A(1\colon k,1\colon k)$ is the leading principal submatrix of order $k$ . For example, for $G_4$ with $k = 2$ , $\bigl[\begin{smallmatrix}1 & 1\\ 1& 2\end{smallmatrix}\bigr]= \bigl[\begin{smallmatrix}1 & 0\\ 1& 1\end{smallmatrix}\bigr] \bigl[\begin{smallmatrix}1 & 1\\ 0& 1\end{smallmatrix}\bigr]$ .

Inverting the Cholesky equation gives $A^{-1} = R^{-1} R^{-T}$ , which implies the interesting relation that the $(n,n)$ element of $A^{-1}$ is $r_{nn}^{-2}$ . So $G_4^{-1}$ has $(4,4)$ element $1/2$ . We also have $\det(A) = \det(R)^2 = (r_{11}\dots r_{nn})^2$ , so $\det(G_4) = 4$ for this matrix.

The Cholesky factorization is named after André-Louis Cholesky (1875–1918), a French military officer involved in geodesy and surveying in Crete and North Africa, who developed it for solving the normal equations arising in least squares problems.

The existence and uniqueness of the factorization can be proved by induction on $n$ , and the proof also serves as a method for computing it. Partition $A\in\mathbb{R}^{n\times n}$ as

$A = \begin{bmatrix} A_{n-1} & c \\ c^T & \alpha \end{bmatrix}.$

We know that $A_{n-1}$ is positive definite (any principal submatrix of a positive definite matrix is easily shown to be positive definite). Assume that $A_{n-1}$ has a unique Cholesky factorization $A_{n-1} = R_{n-1}^TR_{n-1}$ and define the upper triangular matrix

$R = \begin{bmatrix} R_{n-1} & r \\ 0 & \beta \end{bmatrix}.$

Then

$R^T\!R = \begin{bmatrix} R_{n-1}^T R_{n-1} & R_{n-1}^T r \\ r^T R_{n-1} & r^Tr + \beta^2 \end{bmatrix},$

which equals $A$ if and only if

$\begin{aligned} R_{n-1}^T r &= c, \\ r^Tr + \beta^2 &= \alpha. \end{aligned}$

The first equation has a unique solution since $R_{n-1}$ is nonsingular. Then the second equation gives $\beta^2 = \alpha - r^Tr$ . It remains to check that there is a unique real, positive $\beta$ satisfying this equation. From the inequality

$0 < \det(A) = \det(R^T) \det(R) = \det(R_{n-1})^2 \beta^2$

we see that $\beta^2>0$ , hence there is a unique $\beta>0$ . This completes the inductive step.

The quantity $\alpha - r^Tr = \alpha - c^T A_{n-1}^{-1}c$ is the Schur complement of $A_{n-1}$ in $A$ . The essential reason why Cholesky factorization works is that the Schur complements of a positive definite matrix are themselves positive definite. Indeed we have the congruence

$\begin{bmatrix} I & 0 \\ -c^T A_{n-1}^{-1} & 1 \end{bmatrix} \begin{bmatrix} A_{n-1} & c \\ c^T & \alpha \end{bmatrix} \begin{bmatrix} I & -A_{n-1}^{-1}c\\ 0 & 1 \end{bmatrix} = \begin{bmatrix} A_{n-1} & 0 \\ 0 & \alpha - c^T A_{n-1}^{-1} c \end{bmatrix},$

and since congruences preserve definiteness it follows that $\alpha - c^T A_{n-1}^{-1} c > 0$ .

In textbooks it is common to see element-level equations for computing the Cholesky factorization, which come from directly solving the matrix equation $A = R^T\!R$ . If we equate elements on both sides, taking them a column at a time, starting with $a_{11} = r_{11}^2$ , then the following algorithm is obtained:

$\begin{array}{l} 1~ \mbox{for}~j=1:n \\ 2~ \quad \mbox{for}~ i=1:j-1 \\ 3~ \qquad r_{ij} = \bigl(a_{ij} - \sum_{k=1}^{i-1} r_{ki}r_{kj} \bigr) /r_{ii} \\ 4~ \quad \mbox{end} \\ 5~ \quad r_{jj} = \bigl(a_{jj} - \sum_{k=1}^{j-1} r_{kj}^2\bigr)^{1/2} \\ 6~ \mbox{end} \end{array}$

What happens if this algorithm is executed on a general (indefinite) symmetric matrix that is, one that has both positive and negative eigenvalues? Line 5 will either attempt to take the square root of a negative number for some $j$ or it will produce $r_{jj} = 0$ and on the next iteration of the loop we will have a division by zero. In floating-point arithmetic it is possible for the algorithm to fail for a positive definite matrix, but only if it is numerically singular: the algorithm is guaranteed to run to completion if the condition number $\kappa_2(A) = \|A\|_2 \|A^{-1}\|_2$ is safely less than $u^{-1}$ , where $u$ is the unit roundoff.

The MATLAB function chol normally returns an error message if the factorization fails. But a second output argument can be requested, which is set to the number of the stage on which the factorization failed, or to zero if the factorization succeeded. In the case of failure, the partially computed $R$ factor, returned in the first argument, can be used to compute a direction of negative curvature for $A$ , which is a vector $z$ such that $z^T\!Az < 0$ . Here is an example. The code

n = 8;
A = gallery('lehmer',n) - 0.3*eye(n); % Indefinite matrix.
[R,p] = chol(A)
z = [-R\(R'\A(1:p-1,p)); 1; zeros(n-p,1)];
neg_curve = z'*A*z

produces the output

R =
   8.3666e-01   5.9761e-01   3.9841e-01
            0   5.8554e-01   7.3193e-01
            0            0   7.4536e-02
p =
     4
neg_curve =
  -9.1437e+00

Cholesky factorization has excellent numerical stability. The computed factor $\widehat{R}$ satisfies

$A + \Delta A = \widehat{R}^T\widehat{R}, \quad \|\Delta A\|_2 \le c_nu\|A\|_2,$

where $c_n$ is a constant. Unlike for LU factorization there is no possibility of element growth; indeed for $i\le j$ ,

$r_{ij}^2 \le \displaystyle \sum_{k=1}^j r_{kj}^2 = a_{ij},$

so the elements of $R$ are nicely bounded relative to those of $A$ .

Once we have a Cholesky factorization we can use it to solve a linear system $Ax = b$ , by solving the lower triangular system $R^Ty = b$ and then the upper triangular system $Rx = y$ . The computed solution $\widehat{x}$ can be shown to satisfy

$(A+\Delta A)\widehat{x} = b, \quad \|\Delta A\|_2 \le d_nu\|A\|_2$

where $d_n$ is another constant. Thus $\widehat{x}$ has a small backward error.

Finally, what if $A$ is only symmetric positive semidefinite, that is, $x^T\!Ax \ge 0$ for all $x$ , so that $A$ is possibly singular? There always exists a Cholesky factorization (we can take the $R$ factor in the QR factorization of the positive semidefinite square root of $A$ ) but it may not be unique. For example, we have

$A = \left[\begin{array}{cccc} 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 2 & 2\\ 1 & 1 & 2 & 4\\ \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 1 & 1 & 0 & 0\\ 1 & 1 & x & y\\ \end{array}\right] \left[\begin{array}{cccc} 1 & 1 & 1 & 1\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & x\\ 0 & 0 & 0 & y\\ \end{array}\right] = R^T\!R$

for any $x$ and $y$ such that $x^2 + y^2 = 2$ . Note that $A$ has rank 3 but $R$ has two zero diagonal elements. When $A$ is positive semidefinite of rank $r$ what is usually wanted is a Cholesky factorization in which $R$ is zero in its last $n-r$ rows. Such a factorization can be obtained by using complete pivoting, which at each stage permutes the largest remaining diagonal element into the pivot position, which gives a factorization

$\Pi^T\mskip-5mu A\Pi = R^T\!R, \quad R = \begin{bmatrix} R_{11} & R_{12} \\ 0 & 0 \end{bmatrix}, \quad R_{11}\in\mathbb{R}^{r \times r},$

where $\Pi$ is a permutation matrix. For example, with $\Pi$ the identity matrix with its columns in reverse order,

$\Pi^T\mskip-5mu A\Pi = \left[\begin{array}{cccc} 4 & 2 & 1 & 1\\ 2 & 2 & 1 & 1\\ 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \end{array}\right] = \left[\begin{array}{cccc} 2 & 0 & 0 & 0\\ 1 & 1 & 0 & 0\\ \frac{1}{2} & \frac{1}{2} & \frac{\sqrt{2}}{2} & 0\\[2pt] \frac{1}{2} & \frac{1}{2} & \frac{\sqrt{2}}{2} & 0 \end{array}\right] \left[\begin{array}{cccc} 2 & 1 & \frac{1}{2} & \frac{1}{2}\\[2pt] 0 & 1 & \frac{1}{2} & \frac{1}{2}\\[2pt] 0 & 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\\ 0 & 0 & 0 & 0 \end{array}\right],$

which clearly displays the rank of $A$ .

References

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

Claude Brezinski, The Life and Work of André Cholesky, Numer. Algorithms 43, 279–288, 2006.
Nicholas J. Higham, Cholesky factorization, WIREs Comp. Stat. 1(2), 251–254, 2009.
Nicholas J. Higham, Accuracy and Stability of Numerical Algorithms, second edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2002. Chapter 10.

What is Numerical Stability?

Numerical stability concerns how errors introduced during the execution of an algorithm affect the result. It is a property of an algorithm rather than the problem being solved. I will assume that the errors under consideration are rounding errors, but in principle the errors can be from any source.

Consider a scalar function $y = f(x)$ of a scalar $x$ . We regard $x$ as the input data and $y$ as the output. The forward error of a computed approximation $\widehat{y}$ to $y$ is the relative error $|y - \widehat{y}|/|y|$ . The backward error of $\widehat{y}$ is

$\min\left\{\, \displaystyle\frac{|\Delta x|}{|x|}: \widehat{y} = f(x+\Delta x) \,\right\}.$

If $\widehat{y}$ has a small backward error then it is the exact answer for a slightly perturbed input. Here, “small’ is interpreted relative to the floating-point arithmetic, so a small backward error means one of the form $cu$ for a modest constant $c$ , where $u$ is the unit roundoff.

An algorithm that always produces a small backward error is called backward stable. In a backward stable algorithm the errors introduced during the algorithm have the same effect as a small perturbation in the data. If the backward error is the same size as any uncertainty in the data then the algorithm produces as good a result as we can expect.

If $x$ undergoes a relative perturbation of size $u$ then $y = f(x)$ can change by as much as $\mathrm{cond}_f(x) u$ , where

$\mathrm{cond}_f(x) = \lim_{\epsilon\to0} \displaystyle\sup_{|\Delta x| \le \epsilon |x|} \displaystyle\frac{|f(x+\Delta x) - f(x)|}{\epsilon|f(x)|}$

is the condition number of $f$ at $x$ . An algorithm that always produces $\widehat{y}$ with a forward error of order $\mathrm{cond}_f(x) u$ is called forward stable.

The definition of $\mathrm{cond}_f(x)$ implies that a backward stable algorithm is automatically forward stable. The converse is not true. An example of an algorithm that is forward stable but not backward stable is Gauss–Jordan elimination for solving a linear system.

If $\widehat{y}$ satisfies

$\widehat{y} + \Delta y = f(x+\Delta x), \quad |\Delta y| \le \epsilon |\widehat{y}|, \quad |\Delta x| \le \epsilon |x|,$

with $\epsilon$ small in the sense described above, then the algorithm for computing $y$ is mixed backward–forward stable. Such an algorithm produces almost the right answer for a slightly perturbed input. The following diagram illustrates the previous equation.

With these definitions in hand, we can turn to the meaning of the term numerically stable. Depending on the context, numerical stability can mean that an algorithm is (a) backward stable, (b) forward stable, or (c) mixed backward–forward stable.

For some problems, backward stability is difficult or impossible to achieve, so numerical stability has meaning (b) or (c). For example, let $Z = xy^T$ , where $x$ and $y$ are $n$ -vectors. Backward stability would require the computed $\widehat{Z}$ to satisfy $\widehat{Z} = (x+\Delta x)(y+\Delta y)^T$ for some small $\Delta x$ and $\Delta y$ , meaning that $\widehat{Z}$ is a rank- $1$ matrix. But the computed $\widehat{Z}$ contains $n^2$ independent rounding errors and is very unlikely to have rank $1$ .

We briefly mention some other relevant points.

What is the data? For a linear system $Ax = b$ the data is $A$ or $b$ , or both. For a nonlinear function we need to consider whether problem parameters are data; for example, for $f(x) = \cos(3x^2+2)$ are the $3$ and the $2$ constants or are they data, like $x$ , that can be perturbed?
We have measured perturbations in a relative sense. Absolute errors can also be used.
For problems whose inputs are matrices or vectors we need to use norms or componentwise measures.
Some algorithms are only unconditionally numerically stable, that is, they are numerically stable for some subset of problems. For example, the normal equations method for solving a linear least squares problem is forward stable for problems with a large residual.

References

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

Robert M. Corless and Nicolas Fillion, A Graduate Introduction to Numerical Methods from the Viewpoint of Backward Error Analysis, Springer, New York, 2013. My review of this book.
Nicholas J. Higham, Accuracy and Stability of Numerical Algorithms, second edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2002.

What is the Polar Decomposition?

A polar decomposition of $A\in\mathbb{C}^{m \times n}$ with $m \ge n$ is a factorization $A = UH$ , where $U\in\mathbb{C}^{m \times n}$ has orthonormal columns and $H\in\mathbb{C}^{n \times n}$ is Hermitian positive semidefinite. This decomposition is a generalization of the polar representation $z = r \mathrm{e}^{\mathrm{i}\theta}$ of a complex number, where $H$ corresponds to $r\ge 0$ and $U$ to $\mathrm{e}^{\mathrm{i}\theta}$ . When $A$ is real, $H$ is symmetric positive semidefinite. When $m = n$ , $U$ is a square unitary matrix (orthogonal for real $A$ ).

We have $A^*A = H^*U^*UH = H^2$ , so $H = (A^*\!A)^{1/2}$ , which is the unique positive semidefinite square root of $A^*A$ . When $A$ has full rank, $H$ is nonsingular and $U = AH^{-1}$ is unique, and in this case $U$ can be expressed as

$U = \displaystyle\frac{2}{\pi} A \displaystyle\int_{0}^{\infty} (t^2I+A^*\!A)^{-1}\, \mathrm{d}t.$

An example of a polar decomposition is

$A = \left[\begin{array}{@{}rr} 4 & 0\\ -5 & -3\\ 2 & 6 \end{array}\right] = \sqrt{2}\left[\begin{array}{@{}rr} \frac{1}{2} & -\frac{1}{6}\\[\smallskipamount] -\frac{1}{2} & -\frac{1}{6}\\[\smallskipamount] 0 & \frac{2}{3} \end{array}\right] \cdot \sqrt{2}\left[\begin{array}{@{\,}rr@{}} \frac{9}{2} & \frac{3}{2}\\[\smallskipamount] \frac{3}{2} & \frac{9}{2} \end{array}\right] \equiv UH.$

For an example with a rank-deficient matrix consider

$A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix},$

for which $A^*A = \mathrm{diag}(0,1,1)$ and so $H = \mathrm{diag}(0,1,1)$ . The equation $A = UH$ then implies that

$U = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ \theta & 0 & 0 \end{bmatrix}, \quad |\theta| = 1,$

so $U$ is not unique.

The polar factor $U$ has the important property that it is a closest matrix with orthonormal columns to $A$ in any unitarily invariant norm. Hence the polar decomposition provides an optimal way to orthogonalize a matrix. This method of orthogonalization is used in various applications, including in quantum chemistry, where it is called Löwdin orthogonalization. Most often, though, orthogonalization is done through QR factorization, trading optimality for a faster computation.

An important application of the polar decomposition is to the orthogonal Procrustes problem¹

$\min \bigl\{\, \|A-BW\|_F: W \in \mathbb{C}^{n\times n},\; W^*W = I \,\bigr\},$

where $A,B\in\mathbb{C}^{m\times n}$ and the norm is the Frobenius norm $\|A\|_F^2 = \sum_{i,j} |a_{ij}|^2$ . This problem, which arises in factor analysis and in multidimensional scaling, asks how closely a unitary transformation of $B$ can reproduce $A$ . Any solution is a unitary polar factor of $B^*\!A$ , and there is a unique solution if $B^*\!A$ is nonsingular. Another application of the polar decomposition is in 3D graphics transformations. Here, the matrices are $3\times 3$ and the polar decomposition can be computed by exploiting a relationship with quaternions.

For a square nonsingular matrix $A$ , the unitary polar factor $U$ can be computed by a Newton iteration:

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

The iterates $X_k$ converge quadratically to $U$ . This is just one of many iterations for computing $U$ and much work has been done on the efficient implementation of these iterations.

If $A = P \Sigma Q^*$ is a singular value decomposition (SVD), where $P\in\mathbb{C}^{m\times n}$ has orthonormal columns, $Q\in\mathbb{C}^{n\times n}$ is unitary, and $\Sigma$ is square and diagonal with nonnegative diagonal elements, then

$A = PQ^* \cdot Q \Sigma Q^* \equiv UH,$

where $U$ has orthonormal columns and $H$ is Hermitian positive semidefinite. So a polar decomposition can be constructed from an SVD. The converse is true: if $A = UH$ is a polar decomposition and $H = Q\Sigma Q^*$ is a spectral decomposition ( $Q$ unitary, $D$ diagonal) then $A = (UQ)\Sigma Q^* \equiv P \Sigma Q^*$ is an SVD. This latter relation is the basis of a method for computing the SVD that first computes the polar decomposition by a matrix iteration then computes the eigensystem of $H$ , and which is extremely fast on distributed-memory manycore computers.

The nonuniqueness of the polar decomposition for rank deficient $A$ , and the lack of a satisfactory definition of a polar decomposition for $m < n$ , are overcome in the canonical polar decomposition, defined for any $m$ and $n$ . Here, $A = UH$ with $U$ a partial isometry, $H$ is Hermitian positive semidefinite, and $U^*U = HH^+$ . The superscript “ $+$ ” denotes the Moore–Penrose pseudoinverse and a partial isometry can be characterized as a matrix $U$ for which $U^+ = U^*$ .

Generalizations of the (canonical) polar decomposition have been investigated in which the properties of $U$ and $H$ are defined with respect to a general, possibly indefinite, scalar product.

References

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

Nicholas J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008. (Chapter 8.)
Nicholas Higham, Christian Mehl, and Françoise Tisseur, The canonical generalized polar decomposition, SIAM J. Matrix Anal. Appl. 31(4), 2163–2180, 2010.
Nicholas J. Higham and Vanni Noferini, An algorithm to compute the polar decomposition of a $3 \times 3$ matrix, Numer. Algorithms 73, 349–369, 2016.
Yuji Nakatsukasa and Nicholas J. Higham, Stable and efficient spectral divide and conquer algorithms for the symmetric eigenvalue decomposition and the SVD, SIAM J. Sci. Comput. 35(3), A1325–A1349, 2013.
Dalal Sukkari, Hatem Ltaief, Aniello Esposito, and David Keyes, A QDWH-based SVD software framework on distributed-memory manycore systems, ACM Trans. Math. Software 45, 18:1–18:21, 2019.

Footnotes:

Procrustes: an ancient Greek robber who tied his victims to an iron bed, stretching their legs if too short for it, and lopping them if too long.

What Is a Symmetric Positive Definite Matrix?

A real $n\times n$ matrix $A$ is symmetric positive definite if it is symmetric ( $A$ is equal to its transpose, $A^T$ ) and

$x^T\!Ax > 0 \quad \mbox{for all nonzero vectors}~x.$

By making particular choices of $x$ in this definition we can derive the inequalities

$\begin{alignedat}{2} a_{ii} &>0 \quad&&\mbox{for all}~i,\\ a_{ij} &< \sqrt{ a_{ii} a_{jj} } \quad&&\mbox{for all}~i\ne j. \end{alignedat}$

Satisfying these inequalities is not sufficient for positive definiteness. For example, the matrix

$A = \begin{bmatrix} 1 & 3/4 & 0 \\ 3/4 & 1 & 3/4 \\ 0 & 3/4 & 1 \\ \end{bmatrix}$

satisfies all the inequalities but $x^T\!Ax < 0$ for $x = [1,{}-\!\sqrt{2},~1]^T$ .

A sufficient condition for a symmetric matrix to be positive definite is that it has positive diagonal elements and is diagonally dominant, that is, $a_{ii} > \sum_{j \ne i} |a_{ij}|$ for all $i$ .

The definition requires the positivity of the quadratic form $x^T\!Ax$ . Sometimes this condition can be confirmed from the definition of $A$ . For example, if $A = B^T\!B$ and $B$ has linearly independent columns then $x^T\!Ax = (Bx)^T Bx > 0$ for $x\ne 0$ . Generally, though, this condition is not easy to check.

Two equivalent conditions to $A$ being symmetric positive definite are

every leading principal minor $\det(A_k)$ , where the submatrix $A_k = A(1\colon k, 1 \colon k)$ comprises the intersection of rows and columns $1$ to $k$ , is positive,
the eigenvalues of $A$ are all positive.

The first condition implies, in particular, that $\det(A) > 0$ , which also follows from the second condition since the determinant is the product of the eigenvalues.

Here are some other important properties of symmetric positive definite matrices.

$A^{-1}$ is positive definite.
$A$ has a unique symmetric positive definite square root $X$ , where a square root is a matrix $X$ such that $X^2 = A$ .
$A$ has a unique Cholesky factorization $A = R^T\!R$ , where $R$ is upper triangular with positive diagonal elements.

Sources of positive definite matrices include statistics, since nonsingular correlation matrices and covariance matrices are symmetric positive definite, and finite element and finite difference discretizations of differential equations.

Examples of symmetric positive definite matrices, of which we display only the $4\times 4$ instances, are the Hilbert matrix

$H_4 = \left[\begin{array}{@{\mskip 5mu}c*{3}{@{\mskip 15mu} c}@{\mskip 5mu}} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\[6pt] \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5}\\[6pt] \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6}\\[6pt] \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7}\\[6pt] \end{array}\right],$

the Pascal matrix

$P_4 = \left[\begin{array}{@{\mskip 5mu}c*{3}{@{\mskip 15mu} r}@{\mskip 5mu}} 1 & 1 & 1 & 1\\ 1 & 2 & 3 & 4\\ 1 & 3 & 6 & 10\\ 1 & 4 & 10 & 20 \end{array}\right],$

and minus the second difference matrix, which is the tridiagonal matrix

$S_4 = \left[\begin{array}{@{\mskip 5mu}c*{3}{@{\mskip 15mu} r}@{\mskip 5mu}} 2 & -1 & & \\ -1 & 2 & -1 & \\ & -1 & 2 & -1 \\ & & -1 & 2 \end{array}\right].$

All three of these matrices have the property that $a_{ij}$ is non-decreasing along the diagonals. However, if $A$ is positive definite then so is $P^TAP$ for any permutation matrix $P$ , so any symmetric reordering of the row or columns is possible without changing the definiteness.

A $4\times 4$ symmetric positive definite matrix that was often used as a test matrix in the early days of digital computing is the Wilson matrix

$W = \begin{bmatrix} 5 &7& 6& 5 \\ 7 &10& 8& 7 \\ 6& 8& 10& 9 \\ 5& 7& 9 &10 \end{bmatrix}.$

What is the best way to test numerically whether a symmetric matrix is positive definite? Computing the eigenvalues and checking their positivity is reliable, but slow. The fastest method is to attempt to compute a Cholesky factorization and declare the matrix positivite definite if the factorization succeeds. This is a reliable test even in floating-point arithmetic. If the matrix is not positive definite the factorization typically breaks down in the early stages so and gives a quick negative answer.

Symmetric block matrices

$C = \begin{bmatrix} A & X\\ X^T & B \end{bmatrix}$

often appear in applications. If $A$ is nonsingular then we can write

$\begin{bmatrix} A & X\\ X^T & B \end{bmatrix} = \begin{bmatrix} I & 0\\ X^TA^{-1} & I \end{bmatrix} \begin{bmatrix} A & 0\\ 0 & B-X^TA^{-1}X \end{bmatrix} \begin{bmatrix} I & A^{-1}X\\ 0 & I \end{bmatrix},$

which shows that $C$ is congruent to a block diagonal matrix, which is positive definite when its diagonal blocks are. It follows that $C$ is positive definite if and only if both $A$ and $B - X^TA^{-1}X$ are positive definite. The matrix $B - X^TA^{-1}X$ is called the Schur complement of $A$ in $C$ .

We mention two determinantal inequalities. If the block matrix $C$ above is positive definite then $\det(C) \le \det(A) \det(B)$ (Fischer’s inequality). Applying this inequality recursively gives Hadamard’s inequality for a symmetric positive definite $A$ :

$\det(A) \le a_{11}a_{22} \dots a_{nn},$

with equality if and only if $A$ is diagonal.

Finally, we note that if $x^TAx \ge 0$ for all $x\ne0$ , so that the quadratic form is allowed to be zero, then the symmetric matrix $A$ is called symmetric positive semidefinite. Some, but not all, of the properties above generalize in a natural way. An important difference is that semidefinitness is equivalent to all principal minors, of which there are $2^{n-1}$ , being nonnegative; it is not enough to check the $n$ leading principal minors. Consider, as an example, the matrix

$\begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 0 \end{bmatrix},$

which has leading principal minors $1$ , $0$ , and $0$ and a negative eigenvalue.

A complex $n\times n$ matrix $A$ is Hermitian positive definite if it is Hermitian ( $A$ is equal to its conjugate transpose, $A^*$ ) and $x^*Ax > 0$ for all nonzero vectors $x$ . Everything we have said above generalizes to the complex case.

References

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

Rajendra Bhatia, Positive Definite Matrices, Princeton University Press, Princeton, NJ, USA, 2007.
Nicholas J. Higham, Computing a nearest symmetric positive semidefinite matrix, Linear Algebra Appl. 103, 103–118, 1988. Section 5.
Roger A. Horn and Charles R. Johnson, Matrix Analysis, second edition, Cambridge University Press, 2013. My review of the second edition.

What Is the Growth Factor for Gaussian Elimination?

Gaussian elimination is the process of reducing an $n \times n$ matrix to upper triangular form by elementary row operations. It consists of $n$ stages, in the $k$ th of which multiples of row $k$ are added to later rows to eliminate elements below the diagonal in the $k$ th column. The result of Gaussian elimination (assuming it succeeds) is a factorization $A = LU$ , where $L$ is unit lower triangular (lower triangular with ones on the diagonal) and $U$ is upper triangular. This factorization reduces the solution of a linear system $Ax = b$ to the solution of two triangular systems.

James Wilkinson showed in the early 1960s that the numerical stability of Gaussian elimination depends on the growth factor

$\rho_n = \displaystyle\frac{\max_{i,j,k} |a_{ij}^{(k)}|} {\max_{i,j}|a_{ij}|} \ge 1,$

where the $a_{ij}^{(k)}$ are the elements at the start of the $k$ th stage of Gaussian elimination, with $A = \bigl(a_{ij}^{(1)}\bigr)$ . Specifically, he obtained a bound for the backward error of the computed solution that is proportional to $\rho_nu$ , where $u$ is the unit roundoff of the floating-point arithmetic. Wilkinson’s analysis focused attention on the size of the growth factor.

If no row interchanges are done during the elimination then $\rho_n$ can be arbitrarily large, as is easily seen for $n = 2$ . For some specific types of matrix more can be said.

If $A$ is symmetric positive definite then $\rho_n = 1$ . In this case one would normally use Cholesky factorization instead of LU factorization.
If $A$ is nonsingular and totally nonnegative (every submatrix has nonnegative determinant) then $\rho_n = 1$ .
If $A$ is diagonally dominant by rows or columns then $\rho_n \le 2$ .
If $A$ is complex and its real and imaginary parts are both symmetric positive definite then $\rho_n < 3$ .

The entry in the $(k,k)$ position at the start of stage $k$ of Gaussian elimination is called the pivot. By swapping rows and columns, any element with row and column indices greater than or equal to $k$ can be brought into the pivot position and used as the pivot. A pivoting strategy is a strategy for interchanging rows and columns at the start of each stage in order to choose a good sequence of pivots. For dense matrices the main criterion is that the pivoting strategy keeps the growth factor small, while for sparse matrices minimizing fill-in (a zero entry becoming nonzero) is also important.

Partial Pivoting

Partial pivoting interchanges rows to ensure that the pivot element is the largest in magnitude in its column. Wilkinson showed that partial pivoting ensures $\rho_n \le 2^{n-1}$ and that equality is attained for matrices of the form illustrated for $n = 4$ by

$\left[\begin{array}{@{\mskip3mu}rrrr} 1 & 0 & 0 & 1 \\ -1 & 1 & 0 & 1 \\ -1 & -1 & 1 & 1 \\ -1 & -1 & -1 & 1 \\ \end{array} \right].$

This matrix is a special case of a larger class of matrices for which equality is attained (Higham and Higham, 1989). Exponential growth can occur for matrices arising in practical applications, as shown by Wright for the multiple shooting method for two-point boundary value problems and Foster for a quadrature method for solving Volterra integral equations. Remarkably, though, $\rho_n$ is usually of modest size in practice. Explaining this phenomenon is one of the outstanding problems in numerical analysis.

Rook Pivoting

Rook pivoting interchanges rows and columns to ensure that the pivot element is the largest in magnitude in both its row and its column. Foster has shown that

$\rho_n \le 1.5 n^{\frac{3}{4} \log n}.$

This bound grows much more slowly than the bound for partial pivoting, but it can still be large. In practice, the bound is pessimistic.

Complete Pivoting

Complete pivoting chooses as pivot the largest element in magnitude in the remaining part of the matrix. Wilkinson (1961) showed that

$\rho_n \le n^{1/2} (2\cdot 3^{1/2} \cdots n^{1/(n-1)})^{1/2} \sim c \, n^{1/2} n^{\frac{1}{4}\log n}.$

This bound grows much more slowly than that for rook pivoting, but again it is pessimistic. Wilkinson noted in 1965 that no matrix had been discovered for which $\rho_n > n$ . Much interest has focused on the question of whether growth of $n$ can be exceeded and what is the largest possible growth. For a Hadamard matrix, $\rho_n \ge n$ , but such matrices do not exist for all $n$ . Evidence suggests that the growth factor for complete pivoting on a Hadamard matrix is exactly $n$ , and this has been proved for $n = 12$ and $n = 16$ .

That $\rho_n > n$ is possible was shown in the early 1990s by Gould (in floating-point arithmetic) and Edelman (in exact point arithmetic), through a $13\times 13$ matrix with growth $13.02$ .

Growth Factor Bounds for Any Pivoting Strategy

While much attention has focused on investigating the growth factor for particular pivoting strategies, some results are known that provide growth factor lower bounds for any pivoting strategy. Higham and Higham (1989) obtained a result that implies that for the symmetric orthogonal matrix

$S = \sqrt{ \displaystyle\frac{2}{n+1} } \biggl( \sin\Bigl( \displaystyle\frac{ij\pi}{n+1} \Bigr) \biggr)_{i,j=1}^n$

the lower bound

$\rho_n(S)\ge \displaystyle\frac{n+1}{2}$

holds for any pivoting strategy. Recently, Higham, Higham, and Pranesh (2020) have shown that

$\rho_n(Q) \gtrsim \displaystyle\frac{n}{4\log n}$

with high probability for large $n$ when $Q$ is a random orthogonal matrix from the Haar distribution. They show that the same bound holds for matrices that have only one singular value different from $1$ and have singular vector matrices that are random orthogonal matrices from the Haar distribution. These “randsvd matrices” can be generated as gallery('randsvd',n,kappa,2) in MATLAB.

Experimentation

If you want to experiment with growth factors in MATLAB, you can use the function gep from the Matrix Computation Toolbox. Examples:

>> rng(0); A = randn(1000);
>> [L,U,P,Q,rho] = gep(A,'p'); rho  % Partial pivoting.
rho =
   1.9437e+01
>> [L,U,P,Q,rho] = gep(A,'r'); rho  % Rook pivoting.
rho =
   9.9493e+00
>> [L,U,P,Q,rho] = gep(A,'c'); rho  % Complete pivoting.
rho =
   6.8363e+00
>> A = gallery('randsvd',1000,1e1,2);
>> [L,U,P,Q,rho] = gep(A,'c'); rho  % Complete pivoting.
rho =
   5.4720e+01

References

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

Alan Edelman, The complete pivoting conjecture for Gaussian elimination is false, The Mathematica Journal 2, 58–61, 1992.
Leslie V. Foster, The growth factor and efficiency of Gaussian elimination with rook pivoting, J. Comput. Appl. Math. 86, 177–194, 1997
Nicholas J. Higham and Desmond J. Higham, Large growth factors in Gaussian elimination with pivoting, SIAM J. Matrix Anal. Appl. 10 (2), 155–164, 1989.
Desmond J. Higham, Nicholas J. Higham, and Srikara Pranesh, Random Matrices Generating Large Growth in LU Factorization with Pivoting, MIMS EPrint 2020.13, The University of Manchester, UK, May 2020.
Nicholas J. Higham, Accuracy and Stability of Numerical Algorithms, second edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2002. Sections 9.4, 10.4.
Christos Kravvaritis and Marilena Mitrouli, The growth factor of a Hadamard matrix of order $16$ is $16$ , Numer. Linear Algebra Appl. 16, 715–743, 2009.
J. H. Wilkinson, Error analysis of direct methods of matrix inversion, J. Assoc. Comput. Mach. 8, 281–330, 1961.

What Is Stochastic Rounding?

In finite precision arithmetic the result of an elementary arithmetic operation does not generally lie in the underlying number system, $F$ , so it must be mapped back into $F$ by the process called rounding. The most common choice is round to nearest, whereby the nearest number in $F$ to the given number is chosen; this is the default in IEEE standard floating-point arithmetic.

Round to nearest is deterministic: given the same number it always produces the same result. A form of rounding that randomly rounds to the next larger or next smaller number was proposed Barnes, Cooke-Yarborough, and Thomas (1951), Forysthe (1959), and Hull and Swenson (1966). Now called stochastic rounding, it comes in two forms. The first form rounds up or down with equal probability $1/2$ .

We focus here on the second form of stochastic rounding, which is more interesting. To describe it we let $x\notin F$ be the given number and let $x_1$ and $x_2$ with $x_1 < x_2$ be the adjacent numbers in $F$ that are the candidates for the result of rounding. The following diagram shows the situation where $x$ lies just beyond the half-way point between $x_1$ and $x_2$ :

We round up to $x_2$ with probability $(x-x_1)/(x_2-x_1)$ and down to $x_1$ with probability $(x_2-x)/(x_2-x_1)$ ; note that these probabilities sum to $1$ . The probability of rounding to $x_1$ and $x_2$ is proportional to one minus the relative distance from $x$ to each number.

If stochastic rounding chooses to round to $x_1$ in the diagram then it has committed a larger error than round to nearest, which rounds to $x_2$ . Indeed if we focus on floating-point arithmetic then the result of the rounding is

$\mathrm{f\kern.2ptl}(x) = x (1+\delta), \quad |\delta|\le 2u,$

where $u$ is the unit roundoff. For round to nearest the same result is true with the smaller bound $|\delta|\le u$ . In addition, certain results that hold for round to nearest, such as $\mathrm{f\kern.2ptl}(x) = -\mathrm{f\kern.2ptl}(-x)$ and $\mathrm{f\kern.2ptl}(x/\sqrt{x^2+y^2}) \le 1$ , can fail for stochastic rounding. What, then, is the benefit of stochastic rounding?

It is not hard to show that the expected value of the result of stochastically rounding $x$ is $x$ itself (this is obvious if $x = (x_1+x_2)/2$ , for example), so the expected error is zero. Hence stochastic rounding maintains, in a statistical sense, some of the information that is discarded by a deterministic rounding scheme. This property leads to some important benefits, as we now explain.

In certain situations, round to nearest produces correlated rounding errors that cause systematic error growth. This can happen when we form the inner product of two long vectors $x$ and $y$ of nonnegative elements. If the elements all lie on $[0,1]$ then clearly the partial sum can grow monotonically as more and more terms are accumulated until at some point all the remaining terms “drop off the end” of the computed sum and do not change it—the sum stagnates. This phenomenon was observed and analyzed by Huskey and Hartree as long ago as 1949 in solving differential equations on the ENIAC. Stochastic rounding avoids stagnation by rounding up rather than down some of the time. The next figure gives an example. Here, $x$ and $y$ are $n$ -vectors sampled from the uniform $[0,1]$ distribution, the inner product is computed in IEEE standard half precision floating-point arithmetic ( $u \approx 4.9\times 10^{-4}$ ), and the backward errors are plotted for increasing $n$ . From about $n = 1000$ , the errors for stochastic rounding (SR, in orange) are smaller than those for round to nearest (RTN, in blue), the latter quickly reaching 1.

The figure is explained by recent results of Connolly, Higham, and Mary (2020). The key property is that the rounding errors generated by stochastic rounding are mean independent (a weaker version of independence). As a consequence, an error bound proportional to $\sqrt{n}u$ can be shown to hold with high probability. With round to nearest we have only the usual worst-case error bound, which is proportional to $nu$ . In the figure above, the solid black line is the standard backward error bound $nu$ while the dashed black line is $\sqrt{n}u$ . In this example the error with round to nearest is not bounded by a multiple of $\sqrt{n}u$ , as the blue curve crosses the dashed black one. The underlying reason is that the rounding errors for round to nearest are correlated (they are all negative beyond a certain point in the sum), whereas those for stochastic rounding are uncorrelated.

Another notable property of stochastic rounding is that the expected value of the computed inner product is equal to the exact inner product. The same is true more generally for matrix–vector and matrix–matrix products and the solution of triangular systems.

Stochastic rounding is proving popular in neural network training and inference when low precision arithmetic is used, because it avoids stagnation.

Implementing stochastic rounding efficiently is not straightforward, as the definition involves both a random number and an accurate value of the result that we are trying to compute. Possibilities include modifying low level arithmetic routines (Hopkins et al., 2020) and exploiting the augmented operations in the latest version of the IEEE standard floating-point arithmetic (Fasi and Mikaitis, 2020).

Hardware is starting to become available that supports stochastic rounding, including the Intel Lohi neuromorphic chip, the Graphcore Intelligence Processing Unit (intended to accelerate machine learning), and the SpiNNaker2 chip.

Stochastic rounding can be done in MATLAB using the chop function written by me and Srikara Pranesh. This function is intended for experimentation rather than efficient computation.

References

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

R. C. M. Barnes, E. H. Cooke-Yarborough, and D. G. A. Thomas, An electronic digital computor using cold cathode counting tubes for storage, Electronic Eng. 23, 286–291, 1951.
Michael P. Connolly, Nicholas J. Higham and Theo Mary, Stochastic Rounding and Its Probabilistic Backward Error Analysis, MIMS EPrint 2020.12, The University of Manchester, UK, April 2020.
Massimiliano Fasi and Mantas Mikaitis, Algorithms for Stochastically Rounded Elementary Arithmetic Operations in IEEE 754 Floating-Point Arithmetic, MIMS EPrint 2020.9, The University of Manchester, UK, February 2020.
George Forsythe, Reprint of a note on rounding-off errors, SIAM Rev. 1(1), 66–67, 1959.
Suyog Gupta and Ankur Agrawal and Kailash Gopalakrishnan and Pritish Narayanan, Deep Learning with Limited Numerical Precision, in Proceedings of the 32nd International Conference on International Conference on Machine Learning – Volume 37, ICML’15, JMLR.org, 2015, pp. 1737–1746,
Michael Hopkins, Mantas Mikaitis, Dave Lester, and Steve Furber, Stochastic rounding and reduced-precision fixed-point arithmetic for solving neural ordinary differential equations, Phil. Trans. R. Soc. A 378 (2166), 1–22, 2020.
T. E. Hull and J. R. Swenson, Tests of probabilistic models for propagation of roundoff errors, Comm. ACM 9 (3), 108–113, 1966.

What Is the Hilbert Matrix?

The Hilbert matrix $H_n = (h_{ij})$ is the $n\times n$ matrix with $h_{ij} = 1/(i+j-1)$ . For example,

It is probably the most famous test matrix and its conditioning and other properties were extensively studied in the early days of digital computing, especially by John Todd.

The Hilbert matrix is symmetric and it is a Hankel matrix (constant along the anti-diagonals). Less obviously, it is symmetric positive definite (all its eigenvalues are positive) and totally positive (every submatrix has positive determinant). Its condition number grows rapidly with $n$ ; indeed for the 2-norm the asymptotic growth rate is $\kappa_2(H_n) \sim \mathrm{e}^{3.5n}$ . An underlying reason for the ill conditioning is that the Hilbert matrix is obtained when least squares polynomial approximation is done using the monomial basis, and this is known to be an ill-conditioned polynomial basis.

The inverse of $H_n$ has integer entries, with $(i,j)$ entry

$(-1)^{i+j} (i+j-1) \displaystyle{ n+i-1 \choose n-j } { n+j-1 \choose n-i } { i+j-2 \choose i-1 }^2.$

For example,

$H_4^{-1} = \left[\begin{array}{rrrr} 16 & -120 & 240 & -140 \\ -120 & 1200 & -2700 & 1680 \\ 240 & -2700 & 6480 & -4200 \\ -140 & 1680 & -4200 & 2800 \\ \end{array}\right].$

The alternating sign pattern is a consequence of total positivity.

The determinant is also known explicitly:

$\det(H_n) = \displaystyle\frac{ (1!\, 2!\, \dots (n-1)!\, )^4 }{ 1!\, 2!\, \dots (2n-1)! } \sim 2^{-2n^2}. % \cite{todd54}$

The Hilbert matrix is infinitely divisible, which means that the matrix with $(i,j)$ element $h_{ij}^t$ is positive semidefinite for all nonnegative real numbers $t$ .

Other interesting matrices can be formed from the Hilbert matrix. The Lotkin matrix, A = gallery('lotkin',n) in MATLAB, is the Hilbert matrix with the first row replaced by $1$ s, making it nonsymmetric. Like the Hilbert matrix, it is ill conditioned and has an inverse with integer entries. For the $n\times n$ matrix with $(i,j)$ entry $1/(i+j)$ , which is a submatrix of $H_{n+1}$ , the largest singular value tends to $\pi$ as $n\to\infty$ , with error decaying slowly as $O(1/\log n)$ , as was shown by Taussky.

The Hilbert matrix is not a good test matrix, for several reasons. First, it is too ill conditioned, being numerically singular in IEEE double precision arithmetic for $n = 12$ . Second, it is too special: the definiteness and total positivity, together with the fact that it is a special case of a Cauchy matrix (which has elements $1/(x_i + y_j)$ ), mean that certain quantities associated with it can be calculated more accurately than for a general positive definite matrix. The third reason why the Hilbert matrix is not a good test matrix is that most of its elements cannot be stored exactly in floating-point arithmetic, so the matrix actually stored is a perturbation of $H_n$ —and since $H_n$ is ill conditioned this means (for example) that the inverse of the stored matrix can be very different from the exact inverse. A way around this might be to work the integer matrix $H_n^{-1}$ , but its elements are exactly representable in double precision only for $n\le 12$ .

MATLAB has functions hilb and invhilb for the Hilbert matrix and its inverse. How to efficiently form the Hilbert matrix in MATLAB is an interesting question. The hilb function essentially uses the code

j = 1:n;
H = 1./(j'+j-1)

This code is more concise and efficient than two nested loops would be. The second line may appear to be syntactically incorrect, because j'+j adds a column vector to a row vector. However, the MATLAB implicit expansion feature expands j' into an $n\times n$ matrix in which every column is j', expands j into an $n\times n$ matrix in which every row is j, then adds the two matrices. It is not hard to see that the Hilbert matrix is the result of these two lines of code.

References

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

Rajendra Bhatia, Infinitely divisible matrices, Amer. Math. Monthly 113(3), 221–235, 2006.
Man-Duen Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly 90(5), 301–312, 1983
Nicholas J. Higham, Accuracy and Stability of Numerical Algorithms, second edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2002. Section 28.1.
M. Lotkin, A set of test matrices, M.T.A.C. 9(52), 153–161, 1955.

What Is a Fréchet Derivative?

Let $U$ and $V$ be Banach spaces (complete normed vector spaces). The Fréchet derivative of a function $f:U \to V$ at $X\in U$ is a linear mapping $L:U\to V$ such that

$\notag f(X+E) - f(X) - L(X,E) = o(\|E\|)$

for all $E\in U$ . The notation $L(X,E)$ should be read as “the Fréchet derivative of $f$ at $X$ in the direction $E$ ”. The Fréchet derivative may not exist, but if it does exist then it is unique. When $U = V = \mathbb{R}$ , the Fréchet derivative is just the usual derivative of a scalar function: $L(x,e) = f'(x)e$ .

As a simple example, consider $U = V = \mathbb{R}^{n\times n}$ and $f(X) = X^2$ . From the expansion

$f(X+E) - f(X) = XE + EX + E^2$

we deduce that $L(X,E) = XE + EX$ , the first order part of the expansion. If $X$ commutes with $E$ then $L_X(E) = 2XE = 2EX$ .

More generally, it can be shown that if $f$ has the power series expansion $f(x) = \sum_{i=0}^\infty a_i x^i$ with radius of convergence $r$ then for $X,E\in\mathbb{R}^{n\times n}$ with $\|X\| < r$ , the Fréchet derivative is

$L(X,E) = \displaystyle\sum_{i=1}^\infty a_i \displaystyle\sum_{j=1}^i X^{j-1} E X^{i-j}.$

An explicit formula for the Fréchet derivative of the matrix exponential, $f(A) = \mathrm{e}^A$ , is

$L(A,E) = \displaystyle\int_0^1 \mathrm{e}^{A(1-s)} E \mathrm{e}^{As} \, ds.$

Like the scalar derivative, the Fréchet derivative satisfies sum and product rules: if $g$ and $h$ are Fréchet differentiable at $A$ then

$\notag \begin{alignedat}{2} f &= \alpha g + \beta h &&\;\Rightarrow\; L_f(A,E) = \alpha L_g(A,E) + \beta L_h(A,E),\\ f &= gh &&\;\Rightarrow\; L_f(A,E) = L_g(A,E) h(A) + g(A) L_h(A,E). \end{alignedat}$

A key requirement of the definition of Fréchet derivative is that $L(X,E)$ must satisfy the defining equation for all $E$ . This is what makes the Fréchet derivative different from the Gâteaux derivative (or directional derivative), which is the mapping $G:U \to V$ given by

$\notag G(X,E) = \lim_{t\to0} \displaystyle\frac{f(X+t E)-f(X)}{t} = \frac{\mathrm{d}}{\mathrm{dt}}\Bigm|_{t=0} f(X + tE).$

Here, the limit only needs to exist in the particular direction $E$ . If the Fréchet derivative exists at $X$ then it is equal to the Gâteaux derivative, but the converse is not true.

A natural definition of condition number of $f$ is

$\mathrm{cond}(f,X) = \lim_{\epsilon\to0} \displaystyle\sup_{\|\Delta X\| \le \epsilon \|X\|} \displaystyle\frac{\|f(X+\Delta X) - f(X)\|}{\epsilon\|f(X)\|},$

and it can be shown that $\mathrm{cond}$ is given in terms of the Fréchet derivative by

$\mathrm{cond}(f,X) = \displaystyle\frac{\|L(X)\| \|X\|}{\|f(X)\|},$

where

$\|L(X)\| = \sup_{Z\ne0}\displaystyle\frac{\|L(X,Z)\|}{\|Z\|}.$

For matrix functions, the Fréchet derivative has a number of interesting properties, one of which is that the eigenvalues of $L(X)$ are the divided differences

$\notag f[\lambda_i,\lambda_j] = \begin{cases} \dfrac{ f(\lambda_i)-f(\lambda_j) }{\lambda_i - \lambda_j}, & \lambda_i\ne\lambda_j, \\ f'(\lambda_i), & \lambda_i=\lambda_j, \end{cases}$

for $1\le i,j \le n$ , where the $\lambda_i$ are the eigenvalues of $X$ . We can check this formula in the case $F(X) = X^2$ . Let $(\lambda,u)$ be an eigenpair of $X$ and $(\mu,v)$ an eigenpair of $X^T$ , so that $Xu = \lambda u$ and $X^Tv = \mu v$ , and let $E = uv^T$ . Then

$L(X,E) = XE + EX = Xuv^T + uv^TX = (\lambda + \mu) uv^T.$

So $uv^T$ is an eigenvector of $L(X)$ with eigenvalue $\lambda + \mu$ . But $f[\lambda,\mu] = (\lambda^2-\mu^2)/(\lambda - \mu) = \lambda + \mu$ (whether or not $\lambda$ and $\mu$ are distinct).

References

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

Kendall Atkinson and Weimin Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Springer-Verlag, New York, 2009. (Section 5.3).
Nicholas J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008. (Chapter 3).
James Ortega and Werner Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000. (Section 3.1).

What Is the Adjugate of a Matrix?

The adjugate of an $n\times n$ matrix $A$ is defined by

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

where $A_{pq}$ denotes the submatrix of $A$ obtained by deleting row $p$ and column $q$ . It is the transposed matrix of cofactors. The adjugate is sometimes called the (classical) adjoint and is sometimes written as $A^\mathrm{A}$ .

The adjugate satisfies

$A \mathrm{adj}(A) = \mathrm{adj}(A) A = \det(A)I,$

so for nonsingular $A$ ,

$\mathrm{adj}(A) = \det(A) A^{-1},$

and this equation is often used to give a formula for $A^{-1}$ in terms of determinants.

Since the adjugate is a scalar multiple of the inverse, we would expect it to share some properties of the inverse. Indeed we have

$\begin{aligned} \mathrm{adj}(AB) &= \mathrm{adj}(B) \mathrm{adj}(A),\\ \mathrm{adj}(A^*) &= \mathrm{adj}(A)^*. \end{aligned}$

These properties can be proved by first assuming the matrices are nonsingular—in which case they follow from properties of the determinant and the inverse—and then using continuity of the entries of $\mathrm{adj}(A)$ and the fact that every matrix is the limit of a sequence of nonsingular matrices. Another property that can be proved in a similar way is

$\mathrm{adj}\bigl(\mathrm{adj}(A)\bigr) = (\det A)^{n-2} A.$

If $A$ has rank $n-2$ or less then $\mathrm{adj}(A)$ is just the zero matrix. Indeed in this case every $(n-1)\times(n-1)$ submatrix of $A$ is singular, so $\det(A_{ji}) \equiv 0$ in the definition of $\mathrm{adj}(A)$ . In particular, $\mathrm{adj}(0) = 0$ and $\mathrm{adj}(xy^*) = 0$ for any rank-1 matrix $xy^*$

Less obviously, if $\mathrm{rank}(A) = n-1$ then $\mathrm{adj}(A)$ has rank 1. This can be seen from the formula below that expresses the adjugate in terms of the singular value decomposition (SVD).

If $x,y\in\mathbb{C}^n$ , then for nonsingular $A$ ,

$\begin{aligned} \det(A + xy^*) &= \det\bigl( A(I + A^{-1}x y^*) \bigr)\\ &= \det(A) \det( I + A^{-1}x y^*)\\ &= \det(A) (1+ y^*A^{-1}x)\\ &= \det(A) + y^* (\det(A)A^{-1})x\\ &= \det(A) + y^* \mathrm{adj}(A)x. \end{aligned}$

Again it follows by continuity that for any $A$ ,

$\det(A + xy^*) = \det(A) + y^* \mathrm{adj}(A)x.$

This expression is useful when we need to deal with a rank-1 perturbation to a singular matrix. A related identity is

$\det\left( \begin{bmatrix} A & x \\ y^* & \alpha \\ \end{bmatrix}\right) = \alpha \det(A) - y^* \mathrm{adj}(A) x.$

Let $A = U\Sigma V^*$ be an SVD, where $U$ and $V$ are unitary and $\Sigma = \mathrm{diag}(\sigma_i)$ , with $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_n \ge 0$ . Then

$\mathrm{adj}(A) = \mathrm{adj}(V^*) \mathrm{adj}(\Sigma) \mathrm{adj}(U) = \mathrm{adj}(V)^* \mathrm{adj}(\Sigma) \mathrm{adj}(U).$

It is easy to see that $D = \mathrm{adj}(\Sigma)$ is diagonal, with

$d_{kk} = \displaystyle\prod_{i=1\atop i\ne k }^n \sigma_i.$

Since $U$ and $V$ are unitary and hence nonsingular,

$\begin{aligned} \mathrm{adj}(A) &= \mathrm{adj}(V)^* \mathrm{adj}(\Sigma) \mathrm{adj}(U)\\ &= \overline{\det(V)} V^{-*} \mathrm{adj}(\Sigma) \det(U) U^{-1}\\ &= \overline{\det(V)} V \mathrm{adj}(\Sigma) \det(U) U^*\\ &= \overline{\det(V)} \det(U) \cdot V \mathrm{adj}(\Sigma) U^*. \end{aligned}$

If $\mathrm{rank}(A) = n-1$ then $\sigma_n = 0$ and so

$\mathrm{adj}(A) = \rho\, \sigma_1 \sigma_2 \cdots \sigma_{n-1}v_n^{} u_n^*,$

where $\rho = \overline{\det(V)} \det(U)$ has modulus $1$ and $v_n$ and $u_n$ are the last columns of $V$ and $U$ , respectively.

The definition of $\mathrm{adj}(A)$ does not provide a good means for computing it, because the determinant computations are prohibitively expensive. The following MATLAB function (which is very similar to the function adjoint in the Symbolic Math Toolbox) uses the SVD formula.

function X = adj(A)
%ADJ     Adjugate of a matrix.
%   X = ADJ(A) computes the adjugate of the square matrix A via
%   the singular value decomposition.

n = length(A);
[U,S,V] = svd(A);
D = zeros(n);
for i=1:n
    d = diag(S);
    d(i) = 1;
    D(i,i) = prod(d);
end
X = conj(det(V))*det(U)*V*D*U';

Note that, in common with most SVD codes, the svd function does not return $\det(U)$ and $\det(V)$ , so we must compute them. This function is numerically stable, as shown by Stewart (1998).

Finally, we note that Richter (1954) and Mirsky (1956) obtained the Frobenius norm bound

$\|\mathrm{adj}(A) \|_F \le \displaystyle\frac{ \|A\|_F^{n-1} }{ n^{(n-2)/2} }.$

References

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

Roger A. Horn and Charles R. Johnson, Matrix Analysis, second edition, Cambridge University Press, 2013. Section 0.8.2.
L. Mirsky, The Norms of Adjugate and Inverse Matrices, Archiv der Mathematik 7(4), 276–277, 1956.
Hans Richter, Bemerkung zur Norm der Inversen einer Matrix, Archiv der Mathematik 5, 447–448, 1954.
G. W. Stewart, On the Adjugate Matrix, Linear Algebra Appl. 283, 151–164, 1998.

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.

Update (August 10, 2020): added the second displayed equation.

What Is a Matrix Function?

If you give an array as the argument to a mathematical function in a programming language or problem solving environment you are likely to receive the result of applying the function to each component of the array. For example, here MATLAB exponentiates the integers from 0 to 3:

>> A = [0 1; 2 3], X = exp(A)
A =
    0     1
    2     3
X =
   1.0000e+00   2.7183e+00
   7.3891e+00   2.0086e+01

When the array represents a matrix this componentwise evaluation is not useful, as it does not obey the rules of matrix algebra. To see what properties we might require, consider the matrix square root. This function is notable historically because Cayley considered it in the 1858 paper in which he introduced matrix algebra.

A square root of an $n\times n$ matrix $A$ is a matrix $X$ such that $X^2 = A$ . Any square root $X$ satisfies

$AX = X^2 X = X X^2 = XA,$

so $X$ commutes with $A$ . Also, $A^* = (X^2)^* = (X^*)^2$ , so $X^*$ is a square root of $A^*$ (here, the superscript $*$ denotes the conjugate transpose).

Furthermore, for any nonsingular matrix $Z$ we have

$(Z^{-1}XZ)^2 = Z^{-1}X^2 Z = Z^{-1}A Z.$

If we choose $Z$ as a matrix that takes $A$ to its Jordan canonical form $J$ then we have $(Z^{-1}XZ)^2 = J$ , so that $Z^{-1}XZ$ is a square root of $J$ , or in other words $X$ can be expressed as $X = Z \sqrt{J} Z^{-1}$ .

Generalizing from these properties of the matrix square root it is reasonable to ask that a function $f(A)$ of a square matrix $A$ satisfies

$f(A)$ commutes with $A$ ,
$f(A^*) = f(A)^*$ ,
$f(A) = Z f(J)Z^{-1}$ , where $A$ has the Jordan canonical form $A = ZJZ^{-1}$ .

(These are of course not the only requirements; indeed $f(A) \equiv I$ satisfies all three conditions.)

Many definitions of $f(A)$ can be given that satisfy these and other natural conditions. We will describe just three definitions (a notable omission is a definition in terms of polynomial interpolation).

Power Series

For a function $f$ that has a power series expansion we can define $f(A)$ by substituting $A$ for the variable. It can be shown that the matrix series will be convergent for matrices whose eigenvalues lie within the radius of convergence of the scalar power series. For example, where $\rho$ denotes the spectral radius. This definition is natural for functions that have a power series expansion, but it is rather limited in its applicability.

$\begin{aligned} \cos(A) &= I - \displaystyle\frac{A^2}{2!} + \frac{A^4}{4!} - \frac{A^6}{6!} + \cdots,\\ \log(I+A) &= A - \displaystyle\frac{A^2}{2} + \frac{A^3}{3} - \frac{A^4}{4} + \cdots, \quad \rho(A)<1, \end{aligned}$

Jordan Canonical Form Definition

If $A$ has the Jordan canonical form

$Z^{-1}AZ = J = \mathrm{diag}(J_1, J_2, \dots, J_p),$

where

$J_k = J_k(\lambda_k) = \begin{bmatrix} \lambda_k & 1 & & \\ & \lambda_k & \ddots & \\ & & \ddots & 1 \\ & & & \lambda_k \end{bmatrix} = \lambda_k I + N \in \mathbb{C}^{m_k\times m_k},$

then

$f(A) := Z f(J) Z^{-1} = Z \mathrm{diag}(f(J_k)) Z^{-1},$

where

$f(J_k) := \begin{bmatrix} f(\lambda_k) & f'(\lambda_k) & \dots & \displaystyle\frac{f^{(m_k-1)}(\lambda_k)}{(m_k-1)!} \\ & f(\lambda_k) & \ddots & \vdots \\ & & \ddots & f'(\lambda_k) \\ & & & f(\lambda_k) \end{bmatrix}.$

Notice that $f(J_k)$ has the derivatives of $f$ along its diagonals in the upper triangle. Write $J_k = \lambda_k I + N$ , where $N$ is zero apart from a superdiagonal of 1s. The formula for $f(J_k)$ is just the Taylor series expansion

$f(J_k) = f(\lambda_k I + N) = f(\lambda_k)I + f'(\lambda_k)N + \displaystyle\frac{f''(\lambda_k)}{2!}N^2 + \cdots + \displaystyle\frac{f^{(m_k-1)}(\lambda_k)}{(m_k-1)!}N^{m_k-1}.$

The series is finite because $N^{m_k} = 0$ : as $N$ is powered up the superdiagonal of 1s moves towards the right-hand corner until it eventually disappears.

An immediate consequence of this formula is that $f(A)$ is defined only if the necessary derivatives exist: for each eigenvalue $\lambda_k$ we need the existence of the derivatives at $\lambda_k$ of order up to one less than the size of the largest Jordan block in which $\lambda_k$ appears.

When $A$ is a diagonalizable matrix the definition simplifies considerably: if $A = ZDZ^{-1}$ with $D = \mathrm{diag}(\lambda_i)$ then $f(A) = Zf(D)Z^{-1} = Z \mathrm{diag}(\lambda_i) Z^{-1}$ .

Cauchy Integral Formula

For a function $f$ analytic on and inside a closed contour $\Gamma$ that encloses the spectrum of $A$ the matrix $f(A)$ can be defined by the Cauchy integral formula

$f(A) := \displaystyle\frac{1}{2\pi \mathrm{i}} \int_\Gamma f(z) (zI-A)^{-1} \,\mathrm{d}z.$

This formula is mathematically elegant and can be used to provide short proofs of certain theoretical results.

Equivalence of Definitions

These and several other definitions turn out to be equivalent under suitable conditions. This was recognized by Rinehart, who wrote in 1955

“There have been proposed in the literature since 1880 eight distinct definitions of a matric function, by Weyr, Sylvester and Buchheim, Giorgi, Cartan, Fantappiè, Cipolla, Schwerdtfeger and Richter … All of the definitions except those of Weyr and Cipolla are essentially equivalent.”

Computing a Matrix Function in MATLAB

In MATLAB, matrix functions are distinguished from componentwise array evaluation by the postfix “m” on a function name. Thus expm, logm, and sqrtm are matrix function counterparts of exp, log, and sqrt. Compare the example at the start of this article with

>> A = [0 1; 2 3], X = expm(A)
A =
    0     1
    2     3
X =
   5.2892e+00   8.4033e+00
   1.6807e+01   3.0499e+01

The MATLAB function funm calculates $f(A)$ for general matrix functions $f$ (subject to some restrictions).

References

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

Nicholas J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.
Roger Horn and Charles Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
R. F. Rinehart, The Equivalence of Definitions of a Matric Function, Amer. Math. Monthly 62(6), 395-414, 1955.

References

Related Blog Posts

Share this:

References

Related Blog Posts

Share this:

References

Related Blog Posts

Footnotes:

Share this:

References

Related Blog Posts

Share this:

Partial Pivoting

Rook Pivoting

Complete Pivoting

Growth Factor Bounds for Any Pivoting Strategy

Experimentation

References

Related Blog Posts

Share this:

References

Related Blog Posts

Share this:

References

Related Blog Posts

Share this:

References

Related Blog Posts

Share this:

References

Share this:

Power Series

Jordan Canonical Form Definition

Cauchy Integral Formula

Equivalence of Definitions

Computing a Matrix Function in MATLAB

References

Related Blog Posts

Share this: