What’s New in MATLAB R2020a and R2020b?

In this post I discuss new features in MATLAB R2020a and R2020b. As usual in this series, I focus on a few of the features most relevant to my work. See the release notes for a detailed list of the many changes in MATLAB and its toolboxes.

Exportgraphics (R2020a)

The exportgraphics function is very useful for saving to a file a tightly cropped version of a figure with the border white instead of gray. Simple usages are

exportgraphics(gca,'image.pdf')
exportgraphics(gca,'image.jpg','Resolution',200)


I have previously used the export_fig function, which is not built into MATLAB but is available from File Exchange; I think I will be using exportgraphics instead from now on.

Svdsketch (R2020b)

The new svdsketch function computes the singular value decomposition (SVD) $USV^T$ of a low rank approximation to a matrix ($U$ and $V$ orthogonal, $S$ diagonal with nonnegative diagonal entries). It is mainly intended for use with matrices that are close to having low rank, as is the case in various applications.

This function uses a randomized algorithm that computes a sketch of the given $m$-by-$n$ matrix $A$, which is essentially a product $Q^TA$, where $Q$ is an orthonormal basis for the product $A\Omega$, where $\Omega$ is a random $n$-by-$k$ matrix. The value of $k$ is chosen automatically to achieve $\|USV^T-A\|_F \le \mathrm{tol}\|A\|_F$, where $\mathrm{tol}$ is a tolerance that defaults to $\epsilon^{1/4}$ and must not be less than $\epsilon^{1/2}$, where $\epsilon$ is the machine epsilon ($2\times 10^{-16}$ for double precision). The algorithm includes a power method iteration that refines the sketch before computing the SVD.

The output of the function is an SVD in which $U$ and $V$ are numerically orthogonal and the singular values in $S$ of size $\mathrm{tol}$ or larger are good approximations to singular values of $A$, but smaller singular values in $S$ may not be good approximations to singular values of $A$.

Here is an example. The code

n = 8; rng(1); 8; A = gallery('randsvd',n,1e8,3);
[U,S,V] = svdsketch(A,1e-3);
rel_res = norm(A-U*S*V')/norm(A)
singular_values = [svd(A) [diag(S); zeros(n-length(S),1)]]


produces the following output, with the exact singular values in the first column and the approximate ones in the second column:

rel_res =
1.9308e-06
singular_values =
1.0000e+00   1.0000e+00
7.1969e-02   7.1969e-02
5.1795e-03   5.1795e-03
3.7276e-04   3.7276e-04
2.6827e-05   2.6827e-05
1.9307e-06            0
1.3895e-07            0
1.0000e-08            0


The approximate singular values are correct down to around $10^{-5}$, which is more than the $10^{-3}$ requested. This is a difficult matrix for svdsketch because there is no clear gap in the singular values of $A$.

The padding property of an axis puts some padding between the axis limits and the surrounding box. The code

x = linspace(0,2*pi,50); plot(x,tan(x),'linewidth',1.4)
title('Original axis')


produces the output

Turbo Colormap (2020b)

The default colormap changed from jet (the rainbow color map) to parula in R2014b (with a tweak in R2017a), because parula is more perceptually uniform and maintains information when printed in monochrome. The new turbo colormap is a more perceptually uniform version of jet, as these examples show. Notice that turbo has a longer transition through the greens and yellows. If you can’t give up on jet, use turbo instead.

Turbo:

Jet:

Parula:

ND Arrays (R2020b)

The new pagemtimes function performs matrix multiplication on pages of $n$-dimensional arrays, while pagetranspose and pagectranspose carry out the transpose and conjugate transpose, respectively, on pages of $n$-dimensional arrays.

Performance

Both releases report significantly improved speed of certain functions, including some of the ODE solvers.

What Is the Singular Value Decomposition?

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

$\notag A = U\Sigma V^T,$

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

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

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

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

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

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

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

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

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

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

We can write the SVD as

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

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

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

and that the minimum is attained at

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

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

Relations with Symmetric Eigenvalue Problem

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

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

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

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

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

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

Connections with Other Problems

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

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

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

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

Applications

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

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

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

History and Computation

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

References

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

Related Blog Posts

Posted in what-is | 1 Comment

What Is the Complex Step Approximation?

In many situations we need to evaluate the derivative of a function but we do not have an explicit formula for the derivative. The complex step approximation approximates the derivative (and the function value itself) from a single function evaluation. The catch is that it involves complex arithmetic.

For an analytic function $f$ we have the Taylor expansion

$\notag \qquad\qquad\qquad\qquad f(x + \mathrm{i}h) = f(x) + \mathrm{i}h f'(x) - h^2\displaystyle\frac{f''(x)}{2} + O(h^3), \qquad\qquad\qquad\qquad(*)$

where $\mathrm{i} = \sqrt{-1}$ is the imaginary unit. Assume that $f$ maps the real line to the real line and that $x$ and $h$ are real. Then equating real and imaginary parts in $(*)$ gives $\mathrm{Re} f(x+\mathrm{i}h) = f(x) + O(h^2)$ and $\mathrm{Im} f(x+\mathrm{i}h) = hf'(x) + O(h^3)$. This means that for small $h$, the approximations

$\notag f(x) \approx \mathrm{Re} f(x+\mathrm{i}h), \quad f'(x) \approx \mathrm{Im} \displaystyle\frac{f(x+\mathrm{i}h)}{h}$

both have error $O(h^2)$. So a single evaluation of $f$ at a complex argument gives, for small $h$, a good approximation to $f'(x)$, as well as a good approximation to $f(x)$ if we need it.

The usual way to approximate derivatives is with finite differences, for example by the forward difference approximation

$\notag f'(x) \approx \displaystyle\frac{f(x+h) - f(x)}{h}.$

This approximation has error $O(h)$ so it is less accurate than the complex step approximation for a given $h$, but more importantly it is prone to numerical cancellation. For small $h$, $f(x+h)$ and $f(x)$ agree to many significant digits and so in floating-point arithmetic the difference approximation suffers a loss of significant digits. Consequently, as $h$ decreases the error in the computed approximation eventually starts to increase. As numerical analysis textbooks explain, the optimal choice of $h$ that balances truncation error and rounding errors is approximately

$\notag h_{\mathrm{opt}} = 2\Bigl|\displaystyle\frac{u f(x)}{f''(x))} \Bigr|^{1/2},$

where $u$ is the unit roundoff. The optimal error is therefore of order $u^{1/2}$.

A simple example illustrate these ideas. For the function $f(x) = \mathrm{e}^x$ with $x = 1$, we plot in the figure below the relative error for the finite difference, in blue, and the relative error for the complex step approximation, in orange, for $h$ ranging from about $10^{-5}$ to $10^{-11}$. The dotted lines show $u$ and $u^{1/2}$. The computations are in double precision ($u \approx 1.1\times 10^{-16}$). The finite difference error decreases with $h$ until it reaches about $h_{\mathrm{opt}} = 2.1\times 10^{-8}$; thereafter the error grows, giving the characteristic V-shaped error curve. The complex step error decreases steadily until it is of order $u$ for $h \approx u^{1/2}$, and for each $h$ it is about the square of the finite difference error, as expected from the theory.

Remarkably, one can take $h$ extremely small in the complex step approximation (e.g., $h = 10^{-100}$) without any ill effects from roundoff.

The complex step approximation carries out a form of approximate automatic differentiation, with the variable $h$ functioning like a symbolic variable that propagates through the computations in the imaginary parts.

The complex step approximation applies to gradient vectors and it can be extended to matrix functions. If $f$ is analytic and maps real $n\times n$ matrices to real $n\times n$ matrices and $A$ and $E$ are real then (Al-Mohy and Higham, 2010)

$\notag L_f(A,E) \approx \mathrm{Im} \displaystyle\frac{f(A+\mathrm{i}hE)}{h},$

where $L_f(A,E)$ is the Fréchet derivative of $f$ at $A$ in the direction $E$. It is important to note that the method used to evaluate $f$ must not itself use complex arithmetic (as methods based on the Schur decomposition do); if it does, then the interaction of those complex terms with the much smaller $\mathrm{i}hE$ term can lead to damaging subtractive cancellation.

The complex step approximation has also been extended to higher derivatives by using “different imaginary units” in different components (Lantoine et al., 2012).

Here are some applications where the complex step approximation has been used.

• Sensitivity analysis in engineering applications (Giles et al., 2003).
• Approximating gradients in deep learning (Goodfellow et al., 2016).
• Approximating the exponential of an operator in option pricing (Ackerer and Filipović, 2019).

Software has been developed for automatically carrying out the complex step method—for example, by Shampine (2007).

The complex step approximation has been rediscovered many times. The earliest published appearance that we are aware of is in a paper by Squire and Trapp (1998), who acknowledge earlier work of Lyness and Moler on the use of complex variables to approximate derivatives.

References

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

Related Blog Posts

Posted in what-is | 1 Comment

What Is the Sherman–Morrison–Woodbury Formula?

When a nonsingular $n\times n$ matrix $A$ is perturbed by a matrix of rank $k$, the inverse also undergoes a rank-$k$ perturbation. More precisely, if $E$ has rank $k$ and $B = A+E$ is nonsingular then the identity $A^{-1} - B^{-1} = A^{-1} (B-A) B^{-1}$ shows that

$\notag \mathrm{rank}(A^{-1} - B^{-1}) = \mathrm{rank}(A^{-1} E B^{-1}) = \mathrm{rank}(E) = k.$

The Sherman–Morrison–Woodbury formula provides an explicit formula for the inverse of the perturbed matrix $B$.

Sherman–Morrison Formula

We will begin with the simpler case of a rank-$1$ perturbation: $B = A + uv^*$, where $u$ and $v$ are $n$-vectors, and we consider first the case where $A = I$. We might expect that $(I + uv^*)^{-1} = I + \theta uv^*$ for some $\theta$ (consider a binomial expansion of the inverse). Multiplying out, we obtain

$\notag (I + uv^*) (I + \theta uv^*) = I + (1 + \theta + \theta v^*u) uv^*,$

so the product equals the identity matrix when $\theta = -1/(1 + v^*u)$. The condition that $I + uv*$ be nonsingular is $v^*u \ne -1$ (as can also be seen from $\det(I + uv^*) = 1 + v^*u$, derived in What Is a Block Matrix?). So

$\notag (I + uv^*)^{-1} = I - \displaystyle\frac{1}{1 + v^*u} uv^*.$

For the general case write $B = A + uv^* = A(I + A^{-1}u v^*)$. Inverting this equation and applying the previous result gives

$\notag (A + uv^*)^{-1} = A^{-1} - \frac{A^{-1} uv^* A^{-1}}{1 + v^* A^{-1} u},$

subject to the nonsingularity condition $v^*A^{-1}x \ne -1$. This is known as the Sherman–Morrison formula. It explicitly identifies the rank-$1$ change to the inverse.

As an example, if we take $u = te_i$ and $v = e_j$ (where $e_k$ is the $k$th column of the identity matrix) then, writing $A^{-1} = (\alpha_{ij})$, we have

$\notag \bigl(A + te_ie_j^*\bigr)^{-1} = A^{-1} - \displaystyle\frac{tA^{-1}e_i e_j^* A^{-1}}{1 + t \alpha_{ji}}.$

The Frobenius norm of the change to $A^{-1}$ is

$\notag \displaystyle\frac{ |t|\, \| A^{-1}e_i\|_2 \| e_j^*A^{-1}\|_2 } {|1 + t \alpha_{ji}|}.$

If $t$ is sufficiently small then this quantity is approximately maximized for $i$ and $j$ such that the product of the norms of $i$th column and $j$th row of $A^{-1}$ is maximized. For an upper triangular matrix $i = n$ and $j = 1$ are likely to give the maximum, which means that the inverse of an upper triangular matrix is likely to be most sensitive to perturbations in the $(n,1)$ element of the matrix. To illustrate, we consider the matrix

$\notag T = \left[\begin{array}{rrrr} 1 & -1 & -2 & -3\\ 0 & 1 & -4 & -5\\ 0 & 0 & 1 & -6\\ 0 & 0 & 0 & 1 \end{array}\right]$

The $(i,j)$ element of the following matrix is $\| T^{-1} - (T + 10^{-3}e_ie_j^*)^{-1} \|_F$:

$\notag \left[\begin{array}{cccc} 0.044 & 0.029 & 0.006 & 0.001 \\ 0.063 & 0.041 & 0.009 & 0.001 \\ 0.322 & 0.212 & 0.044 & 0.007 \\ 2.258 & 1.510 & 0.321 & 0.053 \\ \end{array}\right]$

As our analysis suggests, the $(4,1)$ entry is the most sensitive to perturbation.

Sherman–Morrison–Woodbury Formula

Now consider a perturbation $UV^*$, where $U$ and $V$ are $n\times k$. This perturbation has rank at most $k$, and its rank is $k$ if $U$ and $V$ are both of rank $k$. If $I + V^* A^{-1} U$ is nonsingular then $A + UV^*$ is nonsingular and

$\notag (A + UV^*)^{-1} = A^{-1} - A^{-1} U (I + V^* A^{-1} U)^{-1} V^* A^{-1},$

which is the Sherman–Morrison–Woodbury formula. The significance of this formula is that $I + V^* A^{-1} U$ is $k\times k$, so if $k\ll n$ and $A^{-1}$ is known then it is much cheaper to evaluate the right-hand side than to invert $A + UV^*$ directly. In practice, of course, we rarely invert matrices, but rather exploit factorizations of them. If we have an LU factorization of $A$ then we can use it in conjunction with the Sherman–Morrison–Woodbury formula to solve $(A + UV^*)x = b$ in $O(n^2 + k^3)$ flops, as opposed to the $O(n^3)$ flops required to factorize $A + UV^*$ from scratch.

The Sherman–Morrison–Woodbury formula is straightforward to verify, by showing that the product of the two sides is the identity matrix. How can the formula be derived in the first place? Consider any two matrices $F$ and $G$ such that $FG$ and $GF$ are both defined. The associative law for matrix multiplication gives $F(GF) = (FG)F$, or $(I + FG)F = F (I + GF)$, which can be written as $F(I+GF)^{-1} = (I+FG)^{-1}F$. Postmultiplying by $G$ gives

$\notag F(I+GF)^{-1}G = (I+FG)^{-1}FG = (I+FG)^{-1}(I + FG - I) = I - (I+FG)^{-1}.$

Setting $F = U$ and $G = V^*$ gives the special case of the Sherman–Morrison–Woodbury formula with $A = I$, and the general formula follows from $A + UV^* = A(I + A^{-1}U V^*)$.

General Formula

We will give a different derivation of an even more general formula using block matrices. Consider the block matrix

$\notag X = \begin{bmatrix} A & U \\ V^* & -W^{-1} \end{bmatrix}$

where $A$ is $n\times n$, $U$ and $V$ are $n\times k$, and $W$ is $k\times k$. We will obtain a formula for $(A + UWV^*)^{-1}$ by looking at $X^{-1}$.

It is straightforward to verify that

$\notag \begin{bmatrix} A & U \\ V^* & -W^{-1} \end{bmatrix} = \begin{bmatrix} I & 0 \\ V^*A^{-1} & I \end{bmatrix} \begin{bmatrix} A & 0 \\ 0 & -(W^{-1} + V^*A^{-1}U) \end{bmatrix} \begin{bmatrix} I & A^{-1}U \\ 0 & I \end{bmatrix}.$

Hence

\notag \begin{aligned} \begin{bmatrix} A & U \\ V^* & -W^{-1} \end{bmatrix}^{-1} &= \begin{bmatrix} I & -A^{-1}U \\ 0 & I \end{bmatrix}. \begin{bmatrix} A^{-1} & 0 \\ 0 & -(W^{-1} + V^*A^{-1}U)^{-1} \end{bmatrix} \begin{bmatrix} I & 0 \\ -V^*A^{-1} & I \end{bmatrix}\\[\smallskipamount] &= \begin{bmatrix} A^{-1} - A^{-1}U(W^{-1} + V^*A^{-1}U)^{-1}V^*A^{-1} & A^{-1}U(W^{-1} + V^*A^{-1U})^{-1} \\ (W^{-1} + V^*A^{-1}U)^{-1} V^*A^{-1} & -(W^{-1} + V^*A^{-1}U)^{-1} \end{bmatrix}. \end{aligned}

In the $(1,1)$ block we see the right-hand side of a Sherman–Morrison–Woodbury-like formula, but it is not immediately clear how this relates to $(A + UWV^*)^{-1}$. Let $P = \bigl[\begin{smallmatrix} 0 & I \\ I & 0 \end{smallmatrix} \bigr]$, and note that $P^{-1} = P$. Then

$\notag PXP = \begin{bmatrix} -W^{-1} & V^* \\ U & A \end{bmatrix}$

and applying the above formula (appropriately renaming the blocks) gives, with $\times$ denoting a block whose value does not matter,

$\notag PX^{-1}P = (PXP)^{-1} = \begin{bmatrix} \times & \times \\ \times & (A + UWV^*)^{-1} \end{bmatrix}.$

Hence $(X^{-1})_{11} = (A + UWV^*)^{-1}$. Equating our two formulas for $(X^{-1})_{11}$ gives

$\notag \qquad\qquad\qquad (A + UWV^*)^{-1} = A^{-1} - A^{-1}U (W^{-1} + V^*A^{-1}U)^{-1} V^*A^{-1}, \qquad\qquad\qquad(*)$

provided that $W^{-1} + V^*A^{-1}U$ is nonsingular.

To see one reason why this formula is useful, suppose that the matrix $A$ and its perturbation are symmetric and we wish to preserve symmetry in our formulas. The Sherman–Morrison–Woodbury requires us to write the perturbation as $UU^*$, so the perturbation must be positive semidefinite. In $(*)$, however, we can write an arbitrary symmetric perturbation as $UWU^*$, with $W$ symmetric but possibly indefinite, and obtain a symmetric formula.

The matrix $-(W^{-1} + V^*A^{-1}U)$ is the Schur complement of $A$ in $X$. Consequently the inversion formula $(*)$ is intimately connected with the theory of Schur complements. By manipulating the block matrices in different ways it is possible to derive variations of $(*)$. We mention just the simple rewriting

$\notag (A + UWV^*)^{-1} = A^{-1} - A^{-1}U W(I + V^*A^{-1}UW)^{-1} V^*A^{-1},$

which is valid if $W$ is singular, as long as $I + WV^*A^{-1}U$ is nonsingular. Note that the formula is not symmetric when $V = U$ and $W = W^*$. This variant can also be obtained by replacing $U$ by $UW$ in the Sherman–Morrison–Woodbury formula.

Historical Note

Formulas for the change in a matrix inverse under low rank perturbations have a long history. They have been rediscovered on multiple occasions, sometimes appearing without comment within other formulas. Equation $(*)$ is given by Duncan (1944), which is the earliest appearance in print that I am aware of. For discussions of the history of these formulas see Henderson and Searle (1981) or Puntanen and Styan (2005).

References

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

Related Blog Posts

Posted in what-is | 4 Comments

What Is a Block Matrix?

A matrix is a rectangular array of numbers treated as a single object. A block matrix is a matrix whose elements are themselves matrices, which are called submatrices. By allowing a matrix to be viewed at different levels of abstraction, the block matrix viewpoint enables elegant proofs of results and facilitates the development and understanding of numerical algorithms.

A block matrix is defined in terms of a partitioning, which breaks a matrix into contiguous pieces. The most common and important case is for an $n\times n$ matrix to be partitioned as a block $2\times 2$ matrix (two block rows and two block columns). For $n = 4$, partitioning into $2\times 2$ blocks gives

$\notag A = \left[\begin{array}{cc|cc} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\\hline a_{31} & a_{32} & a_{33} & a_{34}\\ a_{41} & a_{42} & a_{43} & a_{44}\\ \end{array}\right] = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix},$

where

$\notag A_{11} = A(1\colon2,1\colon2) = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix},$

and similarly for the other blocks. The diagonal blocks in a partitioning of a square matrix are usually square (but not necessarily so), and they do not have to be of the same dimensions. This same $4\times 4$ matrix could be partitioned as

$\notag A = \left[\begin{array}{c|ccc} a_{11} & a_{12} & a_{13} & a_{14}\\\hline a_{21} & a_{22} & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33} & a_{34}\\ a_{41} & a_{42} & a_{43} & a_{44}\\ \end{array}\right] = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix},$

where $A_{11} = (a_{11})$ is a scalar, $A_{21}$ is a column vector, and $A_{12}$ is a row vector.

The sum $C = A + B$ of two block matrices $A = (A_{ij})$ and $B = (B_{ij})$ of the same dimension is obtained by adding blockwise as long as $A_{ij}$ and $B_{ij}$ have the same dimensions for all $i$ and $j$, and the result has the same block structure: $C_{ij} = A_{ij}+B_{ij}$,

The product $C = AB$ of an $m\times n$ matrix $A = (A_{ij})$ and an $n\times p$ matrix $B = (B_{ij})$ can be computed as $C_{ij} = \sum_k A_{ik}B_{kj}$ as long as the products $A_{ik}B_{kj}$ are all defined. In this case the matrices $A$ and $B$ are said to be conformably partitioned for multiplication. Here, $C$ has as many block rows as $A$ and as many block columns as $B$. For example,

$\notag AB = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} A_{11} B_{11} + A_{12} B_{21} & A_{11} B_{12} + A_{12} B_{22} \\ A_{21} B_{11} + A_{22} B_{21} & A_{21} B_{12} + A_{22} B_{22} \end{bmatrix}$

as long as all the eight products $A_{ik}B_{kj}$ are defined.

Block matrix notation is an essential tool in numerical linear algebra. Here are some examples of its usage.

Matrix Factorization

For an $n\times n$ matrix $A$ with nonzero $(1,1)$ element $\alpha$ we can write

$\notag A = \begin{bmatrix} \alpha & b^T \\ c & D \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ c/\alpha & I_{n-1} \end{bmatrix} \begin{bmatrix} \alpha & b^T \\ 0 & D - cb^T/\alpha \end{bmatrix} = : L_1U_1$

The first row and column of $L_1$ have the correct form for a unit lower triangular matrix and likewise the first row and column of $U_1$ have the correct form for an upper triangular matrix. If we can find an LU factorization $D - cb^T/\alpha = L_2U_2$ of the $(n-1)\times (n-1)$ Schur complement $D$ then $A = L_1\mathrm{diag}(1,L_2)\cdot \mathrm{diag}(1,U_2)U_1$ is an LU factorization of $A$. This construction is the basis of an inductive proof of the existence of an LU factorization (provided all the pivots are nonzero) and it also yields an algorithm for computing it.

The same type of construction applies to other factorizations, such as Cholesky factorization, QR factorization, and the Schur decomposition.

Matrix Inverse

A useful formula for the inverse of a nonsingular block triangular matrix

$\notag T = \begin{bmatrix} T_{11} & T_{12} \\ 0 & T_{22} \end{bmatrix}$

is

$\notag T^{-1} = \begin{bmatrix} T_{11}^{-1} & - T_{11}^{-1}T_{12}T_{22}^{-1}\\ 0 & T_{22}^{-1} \end{bmatrix},$

which has the special case

$\notag \begin{bmatrix} I & X \\ 0 & I \end{bmatrix}^{-1} = \begin{bmatrix} I & -X\\ 0 & I \end{bmatrix}.$

If $T$ is upper triangular then so are $T_{11}$ and $T_{22}$. By taking $T_{11}$ of dimension the nearest integer to $n/2$ this formula can be used to construct a divide and conquer algorithm for computing $T^{-1}$.

We note that $\det(T) = \det(T_{11}) \det(T_{22})$, a fact that will be used in the next section.

Determinantal Formulas

Block matrices provides elegant proofs of many results involving determinants. For example, consider the equations

$\notag \begin{bmatrix} I & -A \\ 0 & I \end{bmatrix} \begin{bmatrix} I+AB & 0 \\ B & I \end{bmatrix} = \begin{bmatrix} I & -A\\ B & I \end{bmatrix} = \begin{bmatrix} I & 0 \\ B & I+BA \end{bmatrix} \begin{bmatrix} I & -A \\ 0 & I \end{bmatrix},$

which hold for any $A$ and $B$ such that $AB$ and $BA$ are defined. Taking determinants gives the formula $\det(I + AB) = \det(I + BA)$. In particular we can take $A = x$, $B = y^T$, for $n$-vectors $x$ and $y$, giving $\det(I + xy^T) = 1 + y^Tx$.

Constructing Matrices with Required Properties

We can sometimes build a matrix with certain desired properties by a block construction. For example, if $X$ is an $n\times n$ involutory matrix ($X^2 = I$) then

$\notag \begin{bmatrix} X & I \\ 0 & -X \end{bmatrix}$

is a (block triangular) $2n\times 2n$ involutory matrix. And if $A$ and $B$ are any two $n\times n$ matrices then

$\notag \begin{bmatrix} I - BA & B \\ 2A-ABA & AB-I \end{bmatrix}$

is involutory.

The Anti Block Diagonal Trick

For $n\times n$ matrices $A$ and $B$ consider the anti block diagonal matrix

$\notag X = \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix}.$

Note that

$\notag X^2 = \begin{bmatrix} AB & 0 \\ 0 & BA \end{bmatrix}, \quad X^{-1} = \begin{bmatrix} 0 & B^{-1} \\ A^{-1} & 0 \end{bmatrix}.$

Using these properties one can show a relation between the matrix sign function and the principal matrix square root:

$\notag \mathrm{sign}\left( \begin{bmatrix} 0 & A \\ I & 0 \end{bmatrix} \right) = \begin{bmatrix} 0 & A^{1/2} \\ A^{-1/2} & 0 \end{bmatrix}.$

This allows one to derive iterations for computing the matrix square root and its inverse from iterations for computing the matrix sign function.

It is easy to derive explicit formulas for all the powers of $X$, and hence for any power series evaluated at $X$. In particular, we have the formula

$\notag \mathrm{e}^X = \left[\begin{array}{cc} \cosh\sqrt{AB} & A (\sqrt{BA})^{-1} \sinh \sqrt{BA} \\[\smallskipamount] B(\sqrt{AB})^{-1} \sinh \sqrt{AB} & \cosh\sqrt{BA} \end{array}\right],$

where $\sqrt{Y}$ denotes any square root of $Y$. With $B = I$, this formula arises in the solution of the ordinary differential equation initial value problem $y'' + Ay = 0$, $y(0)=y_0$, $y'(0)=y'_0$,

The most well known instance of the trick is when $B = A^T$. The eigenvalues of

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

are plus and minus the singular values of $A$, together with $|m-n|$ additional zeros if $A$ is $m\times n$ with $m \ne n$, and the eigenvectors of $X$ and the singular vectors of $A$ are also related. Consequently, by applying results or algorithms for symmetric matrices to $X$ one obtains results or algorithms for the singular value decomposition of $A$.

References

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

• Gene Golub and Charles F. Van Loan, Matrix Computations, fourth edition, Johns Hopkins University Press, Baltimore, MD, USA, 2013.
• Nicholas J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008. (Sections 1.5 and 1.6 for the theory of matrix square roots.)
• Roger A. Horn and Charles R. Johnson, Matrix Analysis, second edition, Cambridge University Press, 2013. My review of the second edition.

What Is a Householder Matrix?

A Householder matrix is an $n\times n$ orthogonal matrix of the form

$\notag P = I - \displaystyle\frac{2}{v^Tv} vv^T, \qquad 0 \ne v \in\mathbb{R}^n.$

It is easily verified that $P$ is

• orthogonal ($P^TP = I$),
• symmetric ($P^T = P$),
• involutory ($P^2 = I$ that is, $P$ is a square root of the identity matrix),

where the last property follows from the first two.

A Householder matrix is a rank-$1$ perturbation of the identity matrix and so all but one of its eigenvalues are $1$. The eigensystem can be fully described as follows.

• $P$ has an eigenvalue $-1$ with eigenvector $v$, since $Pv = -v$.
• $P$ has $n-1$ eigenvalues $1$ with eigenvectors any set of $n-1$ linearly independent vectors orthogonal to $v$, which can be taken to be mutually orthogonal: $Px = x$ for every such $x$.

$P$ has trace $n-2$ and determinant $-1$, as can be derived directly or deduced from the facts that the trace is the sum of the eigenvalues and the determinant is the product of the eigenvalues.

For $n = 2$, a Householder matrix can be written as

$\notag P = \begin{bmatrix} \cos\theta & \sin\theta\\ \sin\theta & -\cos\theta \end{bmatrix}.$

Simple examples of Householder matrices are obtained by choosing $v = e = [1,1,\dots,1]^T$, for which $P = I - (2/n)ee^T$. For $n=2,3,4,5,6$ we obtain the matrices

$\notag \begin{gathered} \left[\begin{array}{@{\mskip2mu}rr@{\mskip2mu}} 0 & -1 \\ -1 & 0 \end{array}\right], \quad \displaystyle\frac{1}{3} \left[\begin{array}{@{\mskip2mu}rrr@{\mskip2mu}} 1 & -2 & -2\\ -2 & 1 & -2\\ -2 & -2 & 1\\ \end{array}\right], \quad \displaystyle\frac{1}{2} \left[\begin{array}{@{\mskip2mu}rrrr@{\mskip2mu}} 1 & -1 & -1 & -1\\ -1 & 1 & -1 & -1\\ -1 & -1 & 1 & -1\\ -1 & -1 & -1 & 1\\ \end{array}\right], \\ \displaystyle\frac{1}{5} \left[\begin{array}{@{\mskip2mu}rrrrr@{\mskip2mu}} 3 & -2 & -2 & -2 & -2\\ -2 & 3 & -2 & -2 & -2\\ -2 & -2 & 3 & -2 & -2\\ -2 & -2 & -2 & 3 & -2\\ -2 & -2 & -2 & -2 & 3 \end{array}\right], \quad \displaystyle\frac{1}{3} \left[\begin{array}{@{\mskip2mu}rrrrrr@{\mskip2mu}} 2 & -1 & -1 & -1 & -1 & -1\\ -1 & 2 & -1 & -1 & -1 & -1\\ -1 & -1 & 2 & -1 & -1 & -1\\ -1 & -1 & -1 & 2 & -1 & -1\\ -1 & -1 & -1 & -1 & 2 & -1\\ -1 & -1 & -1 & -1 & -1 & 2 \end{array}\right]. \end{gathered}$

Note that the $4\times 4$ matrix is $1/2$ times a Hadamard matrix.

Applying $P$ to a vector $x$ gives

$\notag Px = x - \displaystyle\left( \frac{2 v^Tx}{v^Tv} \right) v.$

This equation shows that $P$ reflects $x$ about the hyperplane ${\mathrm{span}}(v)^{\perp}$, as illustrated in the following diagram, which explains why $P$ is sometimes called a Householder reflector. Another way of expressing this property is to write $x = \alpha v + z$, where $z$ is orthogonal to $v$. Then $Px = -\alpha v + z$, so the component of $x$ in the direction $v$ has been reversed. If we take $v = e_i$, the $i$th unit vector, then $P = I - 2e_ie_i^T = \mathrm{diag}(1,1,\dots,-1,1,\dots,1)$, which has $-1$ in the $(i,i)$ position. In this case premultiplying a vector by $P$ flips the sign of the $i$th component.

Transforming a Vector

Householder matrices are powerful tools for introducing zeros into vectors. Suppose we are given vectors $x$ and $y$ and wish to find a Householder matrix $P$ such that $Px=y$. Since $P$ is orthogonal, we require that $\|x\|_2 = \|y\|_2$, and since $P$ can never equal the identity matrix we also require $x \ne y$. Now

$Px = y \quad \Longleftrightarrow \quad x - 2 \left( \displaystyle\frac{v^Tx}{v^Tv} \right) v = y,$

and this last equation has the form $\alpha v = x-y$ for some $\alpha$. But $P$ is independent of the scaling of $v$, so we can set $\alpha=1$. Now with $v=x-y$ we have

$\notag v^Tv = x^Tx + y^Ty -2x^Ty$

and, since $x^Tx = y^Ty$,

$\notag v^Tx = x^Tx - y^Tx = \frac{1}{2} v^Tv.$

Therefore

$\notag Px = x - v = y,$

as required. Most often we choose $y$ to be zero in all but its first component.

Square Roots

What can we say about square roots of a Householder matrix, that is, matrices $X$ such that $X^2 = P$?

We note first that the eigenvalues of $X$ are the square roots of those of $P$ and so $n-1$ of them will be $\pm 1$ and one will be $\pm \mathrm{i}$. This means that $X$ cannot be real, as the nonreal eigenvalues of a real matrix must appear in complex conjugate pairs.

Write $P = I - 2vv^T$, where $v$ is normalized so that $v^Tv = 1$. It is natural to look for a square root of the form $X = I - \theta vv^T$. Setting $X^2 = P$ leads to the quadratic equation $\theta^2-2\theta + 2 = 0$, and hence $\theta = 1 \pm \mathrm{i}$. As expected, these two square roots are complex even though $P$ is real. As an example, $\theta = 1 - \mathrm{i}$ gives the following square root of the matrix above corresponding to $v = e/n^{1/2}$ with $n = 3$:

$\notag X = \displaystyle\frac{1}{3} \left[\begin{array}{@{\mskip2mu}rrr} 2+\mathrm{i} & -1+\mathrm{i} & -1+\mathrm{i}\\ -1+\mathrm{i} & 2+\mathrm{i} & -1+\mathrm{i}\\ -1+\mathrm{i} & -1+\mathrm{i} & 2+\mathrm{i} \end{array}\right].$

A good way to understand all the square roots is to diagonalize $P$, which can be done by a similarity transformation with a Householder matrix! Normalizing $v^Tv = 1$ again, let $w = v - e_1$ and $H = I - 2ww^T/(w^Tw)$. Then from the construction above we know that $Hv = e_1$. Hence

$\notag H^T\!PH = HPH = I - 2 Hv v^T\!H = I - 2 e_1e_1^T = \mathrm{diag}(-1,1,1,\dots,1)=: D.$

Then $P = HDH^T$ and so $X = H \sqrt{D} H^T$ gives $2^n$ square roots on taking all possible combinations of signs on the diagonal for $\sqrt{D}$. Because $P$ has repeated eigenvalues these are not the only square roots. The infinitely many others are obtained by taking non-diagonal square roots of $D$, which are of the form $\mathrm{diag}(\pm i, Y)$, where $Y$ is any non-diagonal square root of the $(n-1)\times (n-1)$ identity matrix, which in particular could be a Householder matrix!

Block Householder Matrix

It is possible to define an $n\times n$ block Householder matrix in terms of a given $Z\in\mathbb{R}^{n\times p}$, where $n\ge p$, as

$\notag P = I - 2 Z(Z^TZ)^+Z^T.$

Here, “$+$” denotes the Moore–Penrose pseudoinverse. For $p=1$, $P$ clearly reduces to a standard Householder matrix. It can be shown that $(Z^TZ)^+Z^T = Z^+$ (this is most easily proved using the SVD), and so

$P = I - 2 ZZ^+ = I - 2 P_Z,$

where $P_Z = ZZ^+$ is the orthogonal projector onto the range of $Z$ (that is, $\mathrm{range}(PZ) = \mathrm{range}(Z)$, $P_Z^2 = P_Z$, and $P_Z = P_Z^T$). Hence, like a standard Householder matrix, $P$ is symmetric, orthogonal, and involutory. Furthermore, premultiplication of a matrix by $P$ has the effect of reversing the component in the range of $Z$.

As an example, here is the block Householder matrix corresponding to $Z = \bigl[\begin{smallmatrix} 1 & 2 & 3 & 4\\ 5 & 6 & 7 & 8 \end{smallmatrix}\bigr]^T$:

$\notag \displaystyle\frac{1}{5} \left[\begin{array}{@{\mskip2mu}rrrr@{\mskip2mu}} -2 & -4 & -1 & 2\\ -4 & 2 & -2 & -1\\ -1 & -2 & 2 & -4\\ 2 & -1 & -4 & -2 \end{array}\right].$

One can show (using the SVD again) that the eigenvalues of $P$ are $-1$ repeated $r$ times and $1$ repeated $n-r$ times, where $r = \mathrm{rank}(Z)$. Hence $\mathrm{trace}(P) = n - 2r$ and $\det(P) = (-1)^r$.

Schreiber and Parlett (1988) note the representation for $n = 2k$,

$\notag P = \pm \mathrm{diag}(Q_1,Q_2) \begin{bmatrix} \cos(2\Theta) & \sin(2\Theta) \\ \sin(2\Theta) & -\cos(2\Theta) \end{bmatrix} \mathrm{diag}(Q_1,Q_2)^T,$

where $Q_1$ and $Q_2$ are orthogonal and $\Theta$ is symmetric positive definite. This formula neatly generalizes the formula for a standard Householder matrix for $n = 2$ given above, and a similar formula holds for odd $n$.

Schreiber and Parlett also show how given $E\in\mathbb{R}^{n\times p}$ ($n > p$) one can construct a block Householder matrix $H$ such that

$\notag HE = \begin{bmatrix} F \\ 0 \end{bmatrix}, \qquad F \in \mathbb{R}^{p\times p}.$

The polar decomposition plays a key role in the theory and algorithms for such $H$.

Rectangular Householder Matrix

We can define a rectangular Householder matrix as follows. Let $m > n$, $u \in \mathbb{R}^n$, $v \in \mathbb{R}^{m-n}$, and

$\notag P = \begin{bmatrix} I_n\\0 \end{bmatrix} + \alpha \begin{bmatrix} u\\v \end{bmatrix}u^T = \begin{bmatrix} I_n + \alpha u u^T\\ \alpha vu^T \end{bmatrix} \in \mathbb{R}^n.$

Then $P^TP = I$, that is, $P$ has orthonormal columns, if

$\alpha = \displaystyle\frac{-2}{u^Tu + v^Tv}.$

Of course, $P$ is just the first $n$ columns of the Householder matrix built from the vector $[u^T~v^T]^T$.

Historical Note

The earliest appearance of Householder matrices is in the book by Turnbull and Aitken (1932). These authors show that if $\|x\|_2 = \|y\|_2$ ($x\ne -y$) then a unitary matrix of the form $R = \alpha zz^* - I$ (in their notation) can be constructed so that $Rx = y$. They use this result to prove the existence of the Schur decomposition. The first systematic use of Householder matrices for computational purposes was by Householder (1958) who used them to construct the QR factorization.

References

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

What is a Sparse Matrix?

A sparse matrix is one with a large number of zero entries. A more practical definition is that a matrix is sparse if the number or distribution of the zero entries makes it worthwhile to avoid storing or operating on the zero entries.

Sparsity is not to be confused with data sparsity, which refers to the situation where, because of redundancy, the data can be efficiently compressed while controlling the loss of information. Data sparsity typically manifests itself in low rank structure, whereas sparsity is solely a property of the pattern of nonzeros.

Important sources of sparse matrices include discretization of partial differential equations, image processing, optimization problems, and networks and graphs. In designing algorithms for sparse matrices we have several aims.

• Store the nonzeros only, in some suitable data structure.
• Avoid operations involving only zeros.
• Preserve sparsity, that is, minimize fill-in (a zero element becoming nonzero).

We wish to achieve these aims without sacrificing speed, stability, or reliability.

An important class of sparse matrices is banded matrices. A matrix $A$ has bandwidth $p$ if the elements outside the main diagonal and the first $p$ superdiagonals and subdiagonals are zero, that is, if $a_{ij} = 0$ for $j>i+p$ and $i>j+p$.

The most common type of banded matrix is a tridiagonal matrix $(p = 1$), of which an archetypal example is the second-difference matrix, illustrated for $n = 5$ by

$\notag A_5 = \left[ \begin{array}{@{}*{4}{r@{\mskip10mu}}r} 2 & -1 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0\\ 0 & -1 & 2 & -1 & 0\\ 0 & 0 &-1 & 2 & -1\\ 0 & 0 & 0 & -1 & 2 \end{array}\right].$

This matrix (or more precisely its negative) corresponds to a centered finite difference approximation to a second derivative: $f''(x) \approx (f(x+h) -2 f(x) + f(x-h))/h^2$.

The following plots show the sparsity patterns for two symmetric positive definite matrices. Here, the nonzero elements are indicated by dots.

The matrices are both from power network problems and they are taken from the SuiteSparse Matrix Collection (https://sparse.tamu.edu/). The matrix names are shown in the titles and the nz values below the $x$-axes are the numbers of nonzeros. The plots were produced using MATLAB code of the form

W = ssget('HB/494_bus'); A = W.A; spy(A)


where the ssget function is provided with the collection. The matrix on the left shows no particular pattern for the nonzero entries, while that on the right has a structure comprising four diagonal blocks with a relatively small number of elements connecting the blocks.

It is important to realize that while the sparsity pattern often reflects the structure of the underlying problem, it is arbitrary in that it will change under row and column reorderings. If we are interested in solving $Ax = b$, for example, then for any permutation matrices $P$ and $Q$ we can form the transformed system $PAQ (Q^*x) = Pb$, which has a coefficient matrix $PAQ$ having permuted rows and columns, a permuted right-hand side $Pb$, and a permuted solution. We usually wish to choose the permutations to minimize the fill-in or (almost equivalently) the number of nonzeros in $L$ and $U$. Various methods have been derived for this task; they are necessarily heuristic because finding the minimum is in general an NP-complete problem. When $A$ is symmetric we take $Q = P^T$ in order to preserve symmetry.

For the HB/494_bus matrix the symmetric reverse Cuthill-McKee permutation gives a reordered matrix with the following sparsity pattern, plotted with the MATLAB commands

r = symrcm(A); spy(A(r,r))


The reordered matrix with a variable band structure that is characteristic of the symmetric reverse Cuthill-McKee permutation. The number of nonzeros is, of course, unchanged by reordering, so what has been gained? The next plots show the Cholesky factors of the HB/494_bus matrix and the reordered matrix. The Cholesky factor for the reordered matrix has a much narrower bandwidth than that for the original matrix and has fewer nonzeros by a factor 3. Reordering has greatly reduced the amount of fill-in that occurs; it leads to a Cholesky factor that is cheaper to compute and requires less storage.

Because Cholesky factorization is numerically stable, the matrix can be permuted without affecting the numerical stability of the computation. For a nonsymmetric problem the choice of row and column interchanges also needs to take into account the need for numerical stability, which complicates matters.

The world of sparse matrix computations is very different from that for dense matrices. In the first place, sparse matrices are not stored as $n\times n$ arrays, but rather just the nonzeros are stored, in some suitable data structure. Programming sparse matrix computations is, consequently, more difficult than for dense matrix computations. A second difference from the dense case is that certain operations are, for practical purposes, forbidden, Most notably, we never invert sparse matrices because of the possibly severe fill-in. Indeed the inverse of a sparse matrix is usually dense. For example, the inverse of the tridiagonal matrix given at the start of this article is

$\notag A_5^{-1} = \displaystyle\frac{1}{6} \begin{bmatrix} 5 & 4 & 3 & 2 & 1\\ 4 & 8 & 6 & 4 & 2\\ 3 & 6 & 9 & 6 & 3\\ 2 & 4 & 6 & 8 & 4\\ 1 & 2 & 3 & 4 & 5 \end{bmatrix}.$

While it is always true that one should not solve $Ax = b$ by forming $x = A^{-1} \times b$, for reasons of cost and numerical stability (unless $A$ is orthogonal!), it is even more true when $A$ is sparse.

Finally, we mention an interesting property of $A_5^{-1}$. Its upper triangle agrees with the upper triangle of the rank-$1$ matrix

$\notag \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{bmatrix} \begin{bmatrix} 5 & 4 & 3 & 2 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 4 & 3 & 2 & 1\\ 10 & 8 & 6 & 4 & 2\\ 15 & 12& 9 & 6 & 3\\ 20 & 16& 12& 8 & 4\\ 25 & 20& 15& 10& 5 \end{bmatrix}.$

This property generalizes to other tridiagonal matrices. So while a tridiagonal matrix is sparse, its inverse is data sparse—as it has to be because in general $A$ depends on $2n-1$ parameters and hence so does $A^{-1}$. One implication of this property is that it is possible to compute the condition number $\kappa_{\infty}(A) = \|A\|_{\infty} \|A^{-1}\|_{\infty}$ of a tridiagonal matrix in $O(n)$ flops.

References

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

What Is the Sylvester Equation?

The Sylvester equation is the linear matrix equation

$AX - XB = C,$

where $A\in\mathbb{C}^{m\times m}$, $B\in\mathbb{C}^{n\times n}$, and $X,C\in\mathbb{C}^{m\times n}$. It is named after James Joseph Sylvester (1814–1897), who considered the homogeneous version of the equation, $AX - XB = 0$ in 1884. Special cases of the equation are $Ax = b$ (a standard linear system), $AX = XA$ (matrix commutativity), $Ax = \lambda x$ (an eigenvalue–eigenvector equation), and $AX = I$ (matrix inversion).

In the case where $B = A$, taking the trace of both sides of the equation gives

$\mathrm{trace}(C) = \mathrm{trace}(AX - XA) = \mathrm{trace}(AX) - \mathrm{trace} (XA) = 0,$

so a solution can exist only when $C$ has zero trace. Hence $AX - XA = I$, for example, has no solution.

To determine when the Sylvester equation has a solution we will transform it into a simpler form. Let $A = URU^*$ and $B = VSV^*$ be Schur decompositions, where $U$ and $V$ are unitary and $R$ and $S$ are upper triangular. Premultiplying the Sylvester equation by $U^*$, postmultiplying by $V$, and setting $Y = U^*XV$ and $D = U^*CV$, we obtain

$RY - YS = D,$

which is a Sylvester equation with upper triangular coefficient matrices. Equating the $j$th columns on both sides leads to

$(R - s_{jj}I) y_j = d_j - \displaystyle\sum_{k=1}^{j-1} s_{kj} y_k, \quad j = 1\colon n.$

As long as the triangular matrices $R - s_{jj}I$ are nonsingular for all $j$ we can uniquely solve for $y_1$, $y_2$, …, $y_n$ in turn. Hence the Sylvester equation has a unique solution if $r_{ii} \ne s_{jj}$ for all $i$ and $j$, that is, if $A$ and $B$ have no eigenvalue in common.

Since the Sylvester is a linear equation it must be possible to express it in the standard form “$Ax = b$”. This can be done by applying the vec operator, which yields

$\qquad\qquad\qquad\qquad\qquad (I_n \otimes A - B^T \otimes I_m) \mathrm{vec}(X) = \mathrm{vec}(C), \qquad\qquad\qquad\qquad\qquad(*)$

where $\otimes$ is the Kronecker product. Using the Schur transformations above it is easy to show that the eigenvalues of the coefficient matrix are given in terms of those of $A$ and $B$ by

$\lambda_{ij} (I_n\otimes A - B^T\otimes I_m) = \lambda_i(A) - \lambda_j(B), \quad i=1\colon m, \quad j=1\colon n,$

so the coefficient matrix is nonsingular when $\lambda_i(A) \ne \lambda_j(B)$ for all $i$ and $j$.

By considering the derivative of $Z(t) = \mathrm{e}^{At}C\mathrm{e}^{-Bt}$, it can be shown that if the eigenvalues of $A$ and $-B$ have negative real parts (that is, $A$ and $-B$ are stable matrices) then

$X = -\displaystyle\int_0^{\infty} \mathrm{e}^{At} C \mathrm{e}^{-Bt} \, \mathrm{d}t$

is the unique solution of $AX - XB = C$.

Applications

An important application of the Sylvester equation is in block diagonalization. Consider the block upper triangular matrix

$T = \begin{bmatrix} A & C\\ 0 & B \end{bmatrix}.$

If we can find a nonsingular matrix $Z$ such that $Z^{-1}TZ = \mathrm{diag}(A,B)$ then certain computations with $T$ become much easier. For example, for any function $f$,

$f(T) = f(Z \mathrm{diag}(A,B) Z^{-1}) = Zf(\mathrm{diag}(A,B)) Z^{-1} = Z\mathrm{diag}(f(A),f(B)) Z^{-1},$

so computing $f(T)$ reduces to computing $f(A)$ and $f(B)$. Setting

$Z = \begin{bmatrix} I & -X\\ 0 & I \end{bmatrix}.$

and noting that $Z^{-1}$ is just $Z$ with the sign of the (1,2) block reversed, we find that

$Z^{-1} TZ = \begin{bmatrix} A & -AX + XB + C\\ 0 & B \end{bmatrix}.$

Hence $Z$ block diagonalizes $T$ if $X$ satisfies the Sylvester equation $AX - XB = C$, which we know is possible if the eigenvalues of $A$ and $B$ are distinct. This restriction is unsurprising, as without it we could use this construction to diagonalize a $2\times 2$ Jordan block, which of course is impossible.

For another way in which Sylvester equations arises consider the expansion $(X+E)^2 = X^2 + XE + EX + E^2$ for square matrices $X$ and $E$, from which it follows that $XE + EX$ is the Fréchet derivative of the function $x^2$ at $X$ in the direction $E$, written $L_{x^2}(X,E)$. Consequently, Newton’s method for the square root requires the solution of Sylvester equations, though in practice certain simplifications can be made to avoid their appearance. We can find the Fréchet derivative of $x^{1/2}$ by applying the chain rule to $\bigl(x^{1/2}\bigr)^2 = x$, which gives $L_{x^2}\left(X^{1/2}, L_{x^{1/2}}(X,E)\right) = E$. Therefore $Z = L_{x^{1/2}}(X,E)$ is the solution to the Sylvester equation $X^{1/2} Z + Z X^{1/2} = E$. Consequently, the Sylvester equation plays a role in the perturbation theory for matrix square roots.

Sylvester equations also arise in the Schur–Parlett algorithm for computing matrix functions, which reduces a matrix to triangular Schur form $T$ and then solves $TF-FT = 0$ for $F = f(T)$, blockwise, by a recurrence.

Solution Methods

How can we solve the Sylvester equation? One possibility is to solve $(*)$ by LU factorization with partial pivoting. However, the coefficient matrix is $mn\times mn$ and LU factorization cannot exploit the Kronecker product structure, so this approach is prohibitively expensive unless $m$ and $n$ are small. It is more efficient to compute Schur decompositions of $A$ and $B$, transform the problem, and solve a sequence of triangular systems, as described above in our derivation of the conditions for the existence of a unique solution. This method was developed by Bartels and Stewart in 1972 and it is implemented in the MATLAB function sylvester.

In recent years research has focused particularly on solving Sylvester equations in which $A$ and $B$ are large and sparse and $C$ has low rank, which arise in applications in control theory and model reduction, for example. In this case it is usually possible to find good low rank approximations to $X$ and iterative methods based on Krylov subspaces have been very successful.

Sensitivity and the Separation

Define the separation of $A$ and $B$ by

$\mathrm{sep}(A,B) = \displaystyle\min_{Z\ne0} \displaystyle\frac{ \|AZ-ZB\|_F }{ \|Z\|_F }.$

The separation is positive if $A$ and $B$ have no eigenvalue in common, which we now assume. If $X$ is the solution to $AX - XB = C$ then

$\notag \mathrm{sep}(A,B) \le \displaystyle\frac{ \|AX-XB\|_F }{ \|X\|_F } = \frac{\|C\|_F}{\|X\|_F},$

so $X$ is bounded by

$\notag \|X\|_F \le \displaystyle\frac{\|C\|_F}{\mathrm{sep}(A,B)}.$

It is not hard to show that $\mathrm{sep}(A,B)^{-1} = \|P^{-1}\|_2$, where $P$ is the matrix in $(*)$. This bound on $\|X\|_F$ is a generalization of $\|x\|_2 \le \|A^{-1}\|_2 \|b\|_2$ for $Ax = b$.

The separation features in a perturbation bound for the Sylvester equation. If

$\notag (A+\Delta A)(X+\Delta X) - (X+\Delta X)(B+\Delta B) = C+\Delta C,$

then

$\notag \displaystyle\frac{ \|\Delta X\|_F }{ \|X\|_F } \le 2\sqrt{3}\, \mathrm{sep}(A,B)^{-1} (\|A\|_F + \|B\|_F) \epsilon + O(\epsilon^2),$

where

$\notag \epsilon = \max \left\{ \displaystyle\frac{\|\Delta A\|_F}{\|A\|_F}, \frac{\|\Delta B\|_F}{\|B\|_F}, \frac{\|\Delta C\|_F}{\|C\|_F} \right\}.$

While we have the upper bound $\mathrm{sep}(A,B) \le \min_{i,j} |\lambda_i(A) - \lambda_j(B)|$, this inequality can be extremely weak for nonnormal matrices, so two matrices can have a small separation even if their eigenvalues are well separated. To illustrate, let $T(\alpha)$ denote the $n\times n$ upper triangular matrix with $\alpha$ on the diagonal and $-1$ in all entries above the diagonal. The following table shows the values of $\mathrm{sep}(T(1),T(1.1))$ for several values of $n$.

Even though the eigenvalues of $A$ and $B$ are $0.1$ apart, the separation is at the level of the unit roundoff for $n$ as small as $8$.

The sep function was originally introduced by Stewart in the 1970s as a tool for studying the sensitivity of invariant subspaces.

Variations and Generalizations

The Sylvester equation has many variations and special cases, including the Lyapunov equation $AX + XA^* = C$, the discrete Sylvester equation $X + AXB = C$, and versions of all these for operators. It has also been generalized to multiple terms and to have coefficient matrices on both sides of $X$, yielding

$\displaystyle\sum_{i=1}^k A_i X B_i = C.$

For $k\le 2$ and $m=n$ this equation can be solved in $O(n^3)$ flops. For $k > 2$, no $O(n^3)$ flops algorithm is known and deriving efficient numerical methods remains an open problem. The equation arises in stochastic finite element discretizations of partial differential equations with random inputs, where the matrices $A_i$ and $B_i$ are large and sparse and, depending on the statistical properties of the random inputs, $k$ can be arbitrarily large.

References

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

Related Blog Posts

Posted in what-is | 1 Comment

What is the Kronecker Product?

The Kronecker product of two matrices $A\in\mathbb{C}^{m\times n}$ and $B\in\mathbb{C}^{p\times q}$ (also called the tensor product) is the $mp\times nq$ matrix1

$A \otimes B = \begin{bmatrix} a_{11}B & a_{12}B & \dots & a_{1n}B \\ a_{21}B & a_{22}B & \dots & a_{2n}B \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}B & a_{m2}B & \dots & a_{mn}B \end{bmatrix}.$

In other words, $A\otimes B$ is the block $m\times n$ matrix with $(i,j)$ block $a_{ij}B$. For example,

$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \otimes \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} = \left[\begin{array}{ccc|ccc} a_{11}b_{11} & a_{11}b_{12} & a_{11}b_{13} & a_{12}b_{11} & a_{12}b_{12} & a_{12}b_{13} \\ a_{11}b_{21} & a_{11}b_{22} & a_{11}b_{23} & a_{12}b_{21} & a_{12}b_{22} & a_{12}b_{23} \\\hline a_{21}b_{11} & a_{21}b_{12} & a_{21}b_{13} & a_{22}b_{11} & a_{22}b_{12} & a_{22}b_{13} \\ a_{21}b_{21} & a_{21}b_{22} & a_{21}b_{23} & a_{22}b_{21} & a_{22}b_{22} & a_{22}b_{23} \end{array}\right].$

Notice that the entries of $A\otimes B$ comprise every possible product $a_{ij}b_{rs}$, which is not the case for the usual matrix product $AB$ when it is defined. Indeed if $A$ and $B$ are $n\times n$ then

• $AB$ is $n\times n$ and contains sums of $n^3$ of the products $a_{ij}b_{rs}$,
• $A\otimes B$ is $n^2\times n^2$ and contains all $n^4$ products $a_{ij}b_{rs}$.

Two key properties of the Kronecker product are

\begin{aligned} (A\otimes B)^* &= A^* \otimes B^*, \label{KP1} \\ (A\otimes B)(C\otimes D)&= AC \otimes BD. \label{KP2} \end{aligned}

The second equality implies that when $x_i$ is an eigenvector of $A\in\mathbb{C}^{m\times m}$ with eigenvalue $\lambda_i$ and $y_j$ is an eigenvector of $B\in\mathbb{C}^{n\times n}$ with eigenvalue $\mu_j$ then

$(A\otimes B)(x_i\otimes y_j) = (Ax_i \otimes By_j) = (\lambda_i x_i \otimes \mu_j y_j) = \lambda_i\mu_j ( x_i \otimes y_j),$

so that $\lambda_i\mu_j$ is an eigenvalue of $A\otimes B$ with eigenvector $x_i \otimes y_j$. In fact, the $mn$ eigenvalues of $A\otimes B$ are precisely $\lambda_i\mu_j$ for $i = 1\colon m$ and $j = 1\colon n$.

Kronecker product structure arises in image deblurring models in which the blur is separable, that is, the blur in the horizontal direction can be separated from the blur in the vertical direction. Kronecker products also arise in the construction of Hadamard matrices. Recall that a Hadamard matrix is a matrix of $\pm1$s whose rows and columns are mutually orthogonal. If $H_n$ is an $n\times n$ Hadamard matrix then

$\begin{bmatrix} \phantom{-} 1 & 1 \\ -1 & 1 \end{bmatrix} \otimes H_n = \begin{bmatrix} \phantom{-} H_n & H_n \\ -H_n & H_n \end{bmatrix}$

is a $2n\times 2n$ Hadamard matrix.

The practical significance of Kronecker product structure is that it allows computations on a large matrix to be reduced to computations on smaller matrices. For example, suppose $A$ and $B$ are Hermitian positive definite matrices and $C = A \otimes B$, which can be shown to be Hermitian positive definite from the properties mentioned above. If $A = R^*R$ and $B = S^*S$ are Cholesky factorizations then

$A \otimes B = (R^*R) \otimes (S^*S) = (R^*\otimes S^*) (R \otimes S) = (R \otimes S)^* (R \otimes S),$

so $R\otimes S$, which is easily seen to be triangular with positive diagonal elements, is the Cholesky factor of $A\otimes B$. If $A$ and $B$ are $n\times n$ then forming $A\otimes B$ and computing its Cholesky factorization costs $O(n^6)$ flops, whereas $R$ and $S$ can be computed in $O(n^3)$ flops. If we want to solve a linear system $(A \otimes B)x = b$ this can be done using $R$ and $S$ without explicitly form $R\otimes S$.

Vec Operator

The vec operator stacks the columns of a matrix into one long vector: if $A = [a_1,a_2,\dots,a_m]$ then $\mathrm{vec}(A) = [a_1^T a_2^T \dots a_m^T]^T$. The vec operator and the Kronecker product interact nicely: for any $A$, $X$, and $B$ for which the product $AXB$ is defined,

$\mathrm{vec}(AXB) = (B^T \otimes A) \mathrm{vec}(X).$

This relation allows us to express a linear system $AXB = C$ in the usual form “$Ax=b$”.

Kronecker Sum

The Kronecker sum of $A\in\mathbb{C}^{m\times m}$ and $B\in\mathbb{C}^{n\times n}$ is defined by $A\oplus B = A \otimes I_n + I_m \otimes B$. The eigenvalues of $A\oplus B$ are $\lambda_{ij} = \lambda_i(A) + \lambda_j(B)$, $i=1\colon m$, $j=1\colon n$, where the $\lambda_i(A)$ are the eigenvalues of $A$ and the $\lambda_j(B)$ are those of $B$.

The Kronecker sum arises when we apply the vec operator to the matrix $AX + XB$:

$\mathrm{vec}(AX + XB) = (I_m \otimes A + B^T \otimes I_n) \mathrm{vec}(X) = (B^T \oplus A) \mathrm{vec}(X).$

Kronecker sum structure also arises in finite difference discretizations of partial differential equations, such as when Poisson’s equation is discretized on a square by the usual five-point operator.

Vec-Permutation Matrix

Since for $A\in\mathbb{C}^{m\times n}$ the vectors $\mathrm{vec}(A)$ and $\mathrm{vec}(A^T)$ contain the same $mn$ elements in different orders, we must have

$\mathrm{vec}(A^T) = \Pi_{m,n} \mathrm{vec}(A),$

for some $mn\times mn$ permutation matrix $\Pi_{m,n}$. This matrix is called the the vec-permutation matrix, and is also known as the commutation matrix.

Kronecker multiplication is not commutative, that is, $A \otimes B \ne B \otimes A$ in general, but $A \otimes B$ and $B \otimes A$ do contain the same elements in different orders. In fact, the two matrices are related by row and column permutations: if $A\in\mathbb{C}^{m\times n}$ and $B\in\mathbb{C}^{p\times q}$ then

$(A\otimes B)\Pi_{n,q} = \Pi_{m,p} (B\otimes A).$

This relation can be obtained as follows: for $X\in\mathbb{C}^{n\times q}$,

\begin{aligned} (B \otimes A)\mathrm{vec}(X) &= \mathrm{vec}(AXB^T)\\ &= \Pi_{p,m} \mathrm{vec}(BX^TA^T)\\ &= \Pi_{p,m} (A\otimes B)\mathrm{vec}(X^T)\\ &= \Pi_{p,m} (A\otimes B) \Pi_{n,q}\mathrm{vec}(X). \end{aligned}

Since these equalities hold for all $X$, we have $B\otimes A = \Pi_{p,m} (A\otimes B) \Pi_{n,q}$, from which the relation follows on using $\Pi_{m,n}\Pi_{n,m} = I$, which can be obtained by replacing $A$ by $A^T$ in the definition of vec.

An explicit expression for the the vec-permutation matrix is

$\Pi_{m,n} = \displaystyle\sum_{i=1}^m \sum_{j=1}^n (e_i^{} e_j^T) \otimes (e_j^{} e_i^T),$

where $e_i$ is the $i$th unit vector.

The following plot shows the sparsity patterns of all the vec permutation matrices $\Pi_{m,n}$ with $mn = 120$, where the title of each subplot is $(m,n)$.

MATLAB

In MATLAB the Kronecker product $A\otimes B$ can be computed as kron(A,B) and $\mathrm{vec}(A)$ is obtained by indexing with a colon: A(:). Be careful using kron as it can generate very large matrices!

Historical Note

The Kronecker product is named after Leopold Kronecker (1823–1891). Henderson et al. (1983) suggest that it should be called the Zehfuss product, after Johann Georg Zehfuss (1832–1891), who obtained the result $\det(A\otimes B) = \det(A)^n \det(B)^m$ for $A\in\mathbb{C}^{m\times m}$ and $B\in\mathbb{C}^{n\times n}$ in 1858.

References

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

Footnotes:

1

The $\otimes$ symbol is typed in $\LaTeX$ as \otimes.

Posted in what-is | 1 Comment

What Is the Gerstenhaber Problem?

When Cayley introduced matrix algebra in 1858, he did much more than merely arrange numbers in a rectangular array. His definitions of addition, multiplication, and inversion produced an algebraic structure that has proved to be immensely useful, and which still holds many mysteries today.

An $n\times n$ matrix has $n^2$ parameters, so the vector space $\mathbb{R}^{n\times n}$ of real matrices has dimension $n^2$, with a basis given by the matrices $E_{ij} = e_i^{}e_j^T$, where $e_i$ is the vector that is zero except for a $1$ in the $i$th entry. In his original paper, Cayley noticed the important property that the powers of a particular matrix $A$ can never span $\mathbb{R}^{n\times n}$. The Cayley–Hamilton theorem says that $A$ satisfies its own characteristic equation, that is, $p(A) = 0$ where $p(t) = \det(t I - A)$ is the characteristic polynomial of $A$. This means that $A^n$ can be expressed as a linear combination of $I$, $A$, …, $A^{n-1}$, so the powers of $A$ span a vector space of dimension at most $n$.

Gerstenhaber proved a generalization of this property in 1961: if $A$ and $B$ are two commuting $n\times n$ matrices then the matrices $A^iB^j$, for all nonnegative $i$ and $j$, generate a vector space of dimension at most $n$. This result is much more difficult to prove than the Cayley–Hamilton theorem. Gerstenhaber’s proof was based on algebraic geometry, but purely matrix-theoretic proofs have been given.

A natural question, called the Gerstenhaber problem, is: does this result extend to three matrices, that is, does the vector space

$S_n = \{\, A^iB^jC^k: 0\le i,j,k \le n-1 \,\}$

have dimension at most $n$ for any $n\times n$ matrices $A$, $B$, and $C$ that commute with each other? (We can limit the powers to $n-1$ by the Cayley–Hamilton theorem.) The problem is defined over any field, but here I focus on the reals.

Before considering the three matrix case let us note that the answer is known to be “no” for four commuting $4\times 4$ matrices $A$, $B$, $C$, and $D$. Indeed let

$A = e_1^{}e_3^T, \quad B = e_1^{}e_4^T, \quad C = e_2^{}e_3^T, \quad D = e_2^{}e_4^T$

and note that all possible products of two of these matrices are zero, so the matrices commute pairwise. Yet $I = A^0, A, B, C, D$ are clearly five linearly independent matrices. Hence Gerstenhaber’s result does not extend to four matrices. It also does not extend to five or more matrices because it is known that the failure of the result for one value of $n$ implies failure for all larger $n$. The question, then, is whether the three matrix case is like the two matrix case or the case for four or more matrices.

A great deal of effort has been put into proving or disproving the Gerstenhaber problem, so far without success. Here are two known facts.

• The result holds for all $n\le 11$.
• By a 1905 result of Schur, the dimension of $S_n$ is at most $1 + \lfloor n^2/4\rfloor$, which is less than $n^2$ but still much bigger than $n$. (Here, $\lfloor \cdot \rfloor$ is the floor function.)

One approach to investigating this problem is to look for a counterexample computationally. For some $n\ge 12$, choose three commuting $n\times n$ matrices $A$, $B$, and $C$, select $m\ge n$ monomials

$X_i = A^{i_p} B^{j_p} C^{k_p}, \quad 1\le p \le m,$

form the matrix

$Y = [\mathrm{vec}(X_1), \mathrm{vec}(X_2), \dots, \mathrm{vec}(X_m)],$

where $\mathrm{vec}$ converts a matrix into a vector by stacking the columns on top of each other, then compute $\mathrm{rank}(Y)$, which is a lower bound on $\mathrm{dim}(S_n)$, and check whether it is greater than $n$.

This simple-minded approach has some obvious difficulties. How do we choose $A$, $B$, and $C$? How do we choose the powers? How do we avoid overflow and underflow and compute a reliable rank, given that we might be dealing with large powers of large matrices?

Holbrook and O’Meara (2020), who have written several papers on the Gerstenhaber problem, which they call the GP problem, state that they “firmly believe the GP will turn out to have a negative answer” and they have developed a sophisticated approach to searching for a case with $\mathrm{dim}(S_n) > n$, which they call a “Eureka”. They first note that $A$, $B$ and $C$ can be assumed to be nilpotent. This means that all three matrices must be defective, because a nondefective nilpotent matrix is zero. Next they note that since commuting matrices are simultaneously unitarily triangularizable, $A$, $B$, and $C$ can be assumed to be strictly upper triangular. Then they note that $A$ can be assumed to be in Weyr canonical form.

The Weyr canonical form is a dual of the Jordan canonical form in which the Jordan matrix is replaced by a Weyr matrix, which is a direct sum of basic Weyr matrices. A basic Weyr matrix has one distinct eigenvalue and is upper block bidiagonal with diagonal blocks that are multiples of the identity matrix and superdiagonal blocks that are rectangular identity matrices. The difference between Jordan and Weyr matrices is illustrated by the example

$J(\lambda) = \left[\begin{array}{ccc|cc} \lambda & 1 & 0 &&\\ 0 & \lambda & 1 &&\\ 0 & 0 & \lambda &&\\\hline & & &\lambda & 1\\ & & & 0 &\lambda \end{array}\right], \quad W(\lambda) = \left[\begin{array}{cc|cc|c} \lambda & 0 & 1 & 0&\\ 0 & \lambda & 0 & 1&\\\hline & & \lambda & 0 &1\\ & & 0 &\lambda & 0\\\hline & & & 0 &\lambda \end{array}\right],$

where the Jordan matrix $J(\lambda)$ and Weyr matrix $W(\lambda)$ are related by $J(\lambda) = P^TW(\lambda)P$ for some permutation matrix $P$. There is an elegant way of relating the block sizes in the Jordan and Weyr matrices via a Young diagram. Form an array whose $k$th column contains $j_k$ dots, where $j_k$ is the size of the $j$th diagonal block of $J$:

$\begin{array}{c|ccc} & j_1 & j_2\\\hline w_1 & \bullet & \bullet\\ w_2 & \bullet & \bullet\\ w_3 & \bullet & \end{array}$

Then the number of dots in the $k$th row is the size of the $k$th diagonal block in the Weyr form.

The reason for using the Weyr form is that whereas any matrix that commutes with a Jordan matrix has Toeplitz blocks, any matrix that commutes with a Weyr matrix is block upper triangular and is uniquely determined by the first block row. By choosing a Weyr form for $A$, commuting matrices $B$ and $C$ can be built up in a systematic way.

Thanks to a result of O’Meara (2020) it suffices to compute modulo a prime $p$, so the computations can be done in exact arithmetic, removing the need to worry about rounding errors or overflow.

Holbrook and O’Meara have mostly tried matrices up to dimension 50, but they feel that “the Loch Ness monster probably lives in deeper water, closer to $100\times 100$”. Their MATLAB codes (40 nicely commented but not optimized M-files) are available on request from the address given in their preprint. If you have the time to spare you might want to experiment with the codes and try to find a Eureka.

References

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