Month: March 2021

Bounds for the Norm of the Inverse of a Triangular Matrix

In many situations we need to estimate or bound the norm of the inverse of a matrix, for example to compute an error bound or to check whether an iterative process is guaranteed to converge. This is the same problem as bounding the condition number \kappa(A) = \|A\| \|A^{-1}\|, assuming \|A\| is easy to compute or estimate. Here, we focus on triangular matrices. The bounds we derive can be applied to a general matrix if an LU or QR factorization is available.

We denote by \|\cdot\| any matrix norm, and we take the consistency condition \|AB\| \le \|A\| \|B\| as one of the defining properties of a matrix norm.

It will be useful to note that

\notag \left[\begin{array}{crrrr} 1 & -\theta & -\theta & -\theta & -\theta\\ & 1 & -\theta & -\theta & -\theta\\ & & 1 & -\theta & -\theta\\ & & & 1 & -\theta\\ & & & & 1 \end{array}\right]^{-1} = \left[\begin{array}{ccccc} 1 & \theta & \theta(1+\theta) & \theta(1+\theta)^2 & \theta(1+\theta)^3\\ & 1 & \theta & \theta(1+\theta) & \theta(1+\theta)^2\\ & & 1 & \theta & \theta(1+\theta)\\ & & & 1 & \theta\\ & & & & 1 \end{array}\right]

and that more generally the inverse of the n\times n upper triangular matrix T(\theta) with

\notag (T(\theta))_{ij} = \begin{cases} 1, & i=j, \\ -\theta, & i<j, \end{cases} \qquad (1)

is given by

\notag \bigl(T(\theta)^{-1}\bigr)_{ij} = \begin{cases} 1, & i=j, \\ \theta(1+\theta)^{j-i-1}, & j > i. \end{cases} \qquad (2)

Lower Bound

First, we consider a general matrix A\in\mathbb{C}^{n\times n} and let \lambda be an eigenvalue with |\lambda| = \rho(A) (the spectral radius) and x a corresponding eigenvector. With X = xe^T \in\mathbb{C}^{n\times n}, where e is the vector of ones, AX = \lambda X, so

\notag |\lambda| \|X\| = \|\lambda X\| = \| AX \| \le \|A\| \|X\|,

which implies |\lambda| \le \|A\| since X\ne 0. Hence

\notag \|A\| \ge \rho(A).

Let T be a triangular matrix. Applying the latter bound to T^{-1}, whose eigenvalues are its diagonal entries t_{ii}^{-1}, gives

\notag \|T^{-1}\| \ge \displaystyle\frac{1}{\min_i |t_{ii}|}. \qquad (3)

Combining this bound with the analogous bound for \|T\| gives

\notag \kappa(T) \ge \displaystyle\frac{\max_i |t_{ii}|}{\min_i |t_{ii}|}. \qquad (4)

We note that commonly used norms satisfy \|A\| \ge \max_{i,j}|a_{ij}|, which yields another proof of (3) and (4).

For any x and y such that y = Tx we have the lower bound \|x\| / \|y\| \le \| T^{-1} \|. We can choose y and then solve the triangular system Tx = y for x to obtain the lower bound. Condition number estimation techniques, which we will describe in another article, provide ways to choose y that usually yield estimates of \| T^{-1} \| correct to within an order of magnitude.

For the 2-norm, we can choose y and then compute x = (T^TT)^{-k}y by repeated triangular solves, obtaining the lower bound (\|x\|_2 / \|y\|_2)^{\frac{1}{2k}} \le \| T^{-1} \|_2. This bound is simply the power method applied to (T^TT)^{-1}.

Upper Bounds

Let T\in\mathbb{C}^{n\times n} be an upper triangular matrix. The upper bounds for \|T^{-1}\| that we will discuss depend only on the absolute values of the elements of T. This limits the ability of the bounds to distinguish between well-conditioned and ill-conditioned matrices. For example, consider

\notag \begin{gathered} T_1 = \left[\begin{array}{crrrr} 1 & -2 & -2 & -2 & -2\\ & 1 & -2 & -2 & -2\\ & & 1 & -2 & -2\\ & & & 1 & -2\\ & & & & 1 \end{array}\right], \quad T_1^{-1} = \left[\begin{array}{ccccc} 1 & 2 & 6 & 18 & 54\\ & 1 & 2 & 6 & 18\\ & & 1 & 2 & 6\\ & & & 1 & 2\\ & & & & 1 \end{array}\right], \\ T_2 = \left[\begin{array}{ccccc} 1 & 2 & 2 & 2 & 2\\ & 1 & 2 & 2 & 2\\ & & 1 & 2 & 2\\ & & & 1 & 2\\ & & & & 1 \end{array}\right], \quad T_2^{-1} = \left[\begin{array}{crrrr} 1 & -2 & 2 & -2 & 2\\ & 1 & -2 & 2 & -2\\ & & 1 & -2 & 2\\ & & & 1 & -2\\ & & & & 1 \end{array}\right]. \end{gathered}

The bounds for T_1^{-1} and T_2^{-1} will be the same, yet the inverses are of different sizes (the more so as the dimension increases).

Let D = \mathrm{diag}(T) and write

\notag T = D(I - N),

where N is strictly upper triangular and hence nilpotent with N^n = 0. Then

\notag T^{-1} = (I + N + N^2 + \cdots + N^{n-1}) D^{-1}.

Taking absolute values and using the triangle inequality gives

\notag |T^{-1}| \le (I + |N| + |N|^2 + \cdots + |N|^{n-1}) |D|^{-1}, \qquad(5)

where the inequalities hold elementwise.

The comparison matrix M(A) associated with a general A\in\mathbb{C}^{n \times n} is the matrix with

\notag (M(A))_{ij} = \begin{cases} |a_{ii}|, & i=j, \\ -|a_{ij}|, & i\ne j. \end{cases}

It is not hard to see that M(T) is upper triangular with M(T) = |D| (I - |N|) and so the bound (5) is

\notag |T^{-1}| \le M(T)^{-1}.

If we replace every element above the diagonal of M(T) by the most negative off-diagonal element in its row we obtain the upper triangular matrix W(T) with

\notag (W(T))_{ij} = \begin{cases} |t_{ii}|, & i=j, \\ -\max_{i+1\le k\le n}|t_{ik}|, & i<j. \\ \end{cases}

Then W(T) = |D| (I - |N_1|), where |N| \le |N_1|, so

\notag \begin{aligned} M(T)^{-1} &= (I + |N| + |N|^2 + \cdots + |N|^{n-1}) |D|^{-1}\\ & \le (I + |N_1| + |N_1|^2 + \cdots + |N_1|^{n-1}) |D|^{-1} = W(T)^{-1}. \end{aligned}

Finally, let Z(T) = \min_i|t_{ii}|(I - |N_2|), where N_2 is strictly upper triangular with every element above the diagonal equal to the maximum element of |N_1|, that is,

\notag (Z(T))_{ij} = \begin{cases} \alpha, & i=j, \\ -\alpha\beta, & i<j, \\ \end{cases} \qquad \alpha = \min_i|t_{ii}|, \quad \beta = \max_{i < j}|t_{ij}|/|t_{ii}|.

Then

\notag \begin{aligned} W(T)^{-1} &= (I + |N_1| + |N_1|^2 + \cdots + |N_1|^{n-1}) |D|^{-1} \\ &\le \alpha^{-1} (I + |N_2| + |N_2|^2 + \cdots + |N_2|^{n-1}) = Z(T)^{-1}. \end{aligned}

We note that M(T), W(T), and Z(T) are all nonsingular M-matrices. We summarize the bounds.

Theorem 1.

If T\in\mathbb{C}^{n\times n} is a nonsingular upper triangular matrix then

\notag |T^{-1}| \le M(T)^{-1} \le W(T)^{-1} \le Z(T)^{-1}. \qquad (6)

We make two remarks.

  • The bounds (6) are equally valid for lower triangular matrices as long as the maxima in the definitions of W(T) and Z(T) are taken over columns instead of rows.
  • We could equally well have written A = (I-N)D. The comparison matrix M(T) = (I - |N|)|D| is unchanged, and (6) continues to hold as long as the maxima in the definitions of W(T) and Z(T) are taken over columns rather than rows.

It follows from the theorem that

\notag \|T^{-1}\| \le \|M(T)^{-1}\| \le \|W(T)^{-1}\| \le \|Z(T)^{-1}\|

for the 1-, 2-, and \infty-norms and the Frobenius norm. Now M(T), W(T), and Z(T) all have nonnegative inverses, and for a matrix A with nonnegative inverse we have \|A^{-1}\|_{\infty} = \|A^{-1}e\|_{\infty}. Hence

\notag \begin{aligned} \|T^{-1}\|_{\infty} &\le \|M(T)^{-1}e\|_{\infty} \le \|W(T)^{-1}e\|_{\infty} \le \|Z(T)^{-1}e\|_{\infty}\\ O(n^3) \hskip10pt & \hskip35pt O(n^2) \hskip65pt O(n) \hskip65pt O(1) \end{aligned}

where the big-Oh expressions show the asymptotic cost in flops of evaluating each term by solving the relevant triangular system. As the bounds become less expensive to compute they become weaker. The quantity \|Z(T)^{-1}\|_p can be explicitly evaluated for p = \infty, using (2). It has the same value for p = 1, and since \|A\|_2 \le (\|A\|_1\|A\|_{\infty})^{1/2} we have

\notag \|T^{-1}\|_p \le \displaystyle\frac{ (\beta + 1)^{n-1}}{\alpha}, \quad p = 1,2,\infty. \qquad(7)

This bound is an equality for p = 1,\infty for the matrix T(\theta) in (1).

For the Frobenius norm, evaluating \|Z(T)^{-1}\|_F, and using \|A\|_2 \le \|A\|_F, gives

\notag \|T^{-1}\|_{2,F} \le \displaystyle\frac{ \bigl( (\beta + 1)^{2n} + 2n(\beta + 2) - 1 \bigr)^{1/2}} {\alpha(\beta + 2)}. \qquad(8)

For the 2-norm, either of (7) and (8) can be the smaller bound depending on \beta.

For the special case of a bidiagonal matrix B it is easy to show that |B^{-1}| = M(B)^{-1}, and so \|B^{-1}\|_{\infty} = \|M(B)^{-1}\|_{\infty} = \|M(B)^{-1}e\|_{\infty} can be computed exactly in O(n) flops.

These upper bounds can be arbitrarily weak, even for a fixed n, as shown by the example

\notag T(\theta) = \begin{bmatrix} \theta^{-1} & 1 & 1 \\ 0 & \theta^{-1} & \theta^{-1} \\ 0 & 0 & \theta^{-2} \end{bmatrix}, \quad \theta > 0,

for which

\notag T(\theta)^{-1} = \begin{bmatrix} \theta & -\theta^2 & 0 \\ 0 & \theta & -\theta^2 \\ 0 & 0 & \theta^2 \end{bmatrix}, \quad M(T(\theta))^{-1} = \begin{bmatrix} \theta & \theta^2 & 2\theta^3 \\ 0 & \theta & \theta^2 \\ 0 & 0 & \theta^2 \end{bmatrix}.

As \theta\to\infty, \|M(T(\theta))^{-1}\|_{\infty} /\|T(\theta)^{-1}\|_{\infty} \approx 2\theta. On the other hand, the overestimation is bounded as a function of n for triangular matrices resulting from certain pivoting strategies.

Theorem 1.

Suppose the upper triangular matrix T\in\mathbb{C}^{n\times n} satisfies

\notag |t_{ii}| \ge |t_{ij}|, \quad j>i. \qquad (9)

Then, for the 1-, 2-, and \infty-norms,

\notag \displaystyle\frac{1}{\min_i|t_{ii}|} \le \|T^{-1}\| \le \|M(T)^{-1}\| \le \|W(T)^{-1}\| \le \|Z(T)^{-1}\| \le \displaystyle\frac{2^{n-1}}{{\min_i|t_{ii}|}}.

Proof. The first four inequalities are a combination of (3) and (6). The fifth inequality is obtained from the expression (7) for \|Z(T)^{-1}\| with \beta = 1.

Condition (9) is satisfied for the triangular factors from QR factorization with column pivoting and for the transpose of the unit lower triangular factors from LU factorization with any form of pivoting.

The upper bounds we have described have been derived independently by several authors, as explained by Higham (2002).

References

What Is a Companion Matrix?

A companion matrix C\in\mathbb{C}^{n\times n} is an upper Hessenberg matrix of the form

\notag C = \begin{bmatrix} a_{n-1} & a_{n-2} & \dots &\dots & a_0 \\ 1 & 0 & \dots &\dots & 0 \\ 0 & 1 & \ddots & & \vdots \\ \vdots & & \ddots & 0 & 0 \\ 0 & \dots & \dots & 1 & 0 \end{bmatrix}.

Alternatively, C can be transposed and permuted so that the coefficients a_i appear in the first or last column or the last row. By expanding the determinant about the first row it can be seen that

\notag \det(\lambda I - C) = \lambda^n - a_{n-1}\lambda^{n-1} - \cdots - a_1\lambda - a_0, \qquad (*)

so the coefficients in the first row of C are the coefficients of its characteristic polynomial. (Alternatively, in \lambda I - C add \lambda^{n-j} times the jth column to the last column for j = 1:n-1, to obtain p(\lambda)e_1 as the new last column, and expand the determinant about the last column.) MacDuffee (1946) introduced the term “companion matrix” as a translation from the German “Begleitmatrix”.

Setting \lambda = 0 in (*) gives \det(C) = (-1)^{n+1} a_0, so C is nonsingular if and only if a_0 \ne 0. The inverse is

\notag C^{-1} = \begin{bmatrix} 0 & 1 & 0 &\dots& 0 \cr 0 & 0 & 1 &\ddots& 0 \cr \vdots & & \ddots & \ddots & 0\cr \vdots & & & \ddots & 1\cr \displaystyle\frac{1}{a_0} & -\displaystyle\frac{a_{n-1}}{a_0} & -\displaystyle\frac{a_{n-2}}{a_0} & \dots & -\displaystyle\frac{a_1}{a_0} \end{bmatrix}.

Note that P^{-1}C^{-1}P is in companion form, where P = I_n(n:-1:1,:) is the reverse identity matrix, and the coefficients are those of the polynomial -\lambda^n p(1/\lambda), whose roots are the reciprocals of those of p.

A companion matrix has some low rank structure. It can be expressed as a unitary matrix plus a rank-1 matrix:

\notag C = \begin{bmatrix} 0 & 0 & \dots &\dots & 1 \\ 1 & 0 & \dots &\dots & 0 \\ 0 & 1 & \ddots & & 0 \\ \vdots & & \ddots & 0 & 0 \\ 0 & \dots & \dots & 1 & 0 \end{bmatrix} + e_1 \begin{bmatrix} a_{n-1} & a_{n-2} & \dots & a_0-1 \end{bmatrix}. \qquad (1)

Also, C^{-T} differs from C in just the first and last columns, so C^{-T} = C + E, where E is a rank-2 matrix.

If \lambda is an eigenvalue of C then [\lambda^{n-1}, \lambda^{n-2}, \dots, \lambda, 1]^T is a corresponding eigenvector. The last n-1 rows of \lambda I - C are clearly linearly independent for any \lambda, which implies that C is nonderogatory, that is, no two Jordan blocks in the Jordan canonical form contain the same eigenvalue. In other words, the characteristic polynomial is the same as the minimal polynomial.

The MATLAB function compan takes as input a vector [p_1,p_2, \dots, p_{n+1}] of the coefficients of a polynomial, p_1x^n + p_2 x^{n-1} + \cdots + p_n x + p_{n+1}, and returns the companion matrix with a_{n-1} = -p_2/p_1, …, a_0 = -p_{n+1}/p_1.

Perhaps surprisingly, the singular values of C have simple representations, found by Kenney and Laub (1988):

\notag \begin{aligned} \sigma_1^2 &= \displaystyle\frac{1}{2} \left( \alpha + \sqrt{\alpha^2 - 4 a_0^2} \right), \\ \sigma_i^2 &= 1, \qquad i=2\colon n-1, \\ \sigma_n^2 &= \displaystyle\frac{1}{2} \left( \alpha - \sqrt{\alpha^2 - 4 a_0^2} \right), \end{aligned}

where \alpha = 1 + a_0^2 + \cdots + a_{n-1}^2. These formulae generalize to block companion matrices, as shown by Higham and Tisseur (2003).

Applications

Companion matrices arise naturally when we convert a high order difference equation or differential equation to first order. For example, consider the Fibonacci numbers 1, 1, 2, 3, 5, \dots, which satisfy the recurrence f_n = f_{n-1} + f_{n-2} for n \ge 2, with f_0 = f_1 = 1. We can write

\notag \begin{bmatrix} f_n \\ f_{n-1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} f_{n-1} \\ f_{n-2} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^2 \begin{bmatrix} f_{n-2} \\ f_{n-3} \end{bmatrix} = \cdots = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{n-1} \begin{bmatrix} f_{1} \\ f_{0} \end{bmatrix},

where \left[\begin{smallmatrix}1 & 1 \\ 1 & 0 \end{smallmatrix}\right] is a companion matrix. This expression can be used to compute f_n in O(\log_2n) operations by computing the matrix power using binary powering.

As another example, consider the differential equation

\notag y''' = b_2 y'' + b_1 y' + b_0 y.

Define new variables

z_1 = y'', \quad z_2 = y', \quad z_3 = y.

Then

\notag \begin{aligned} z_1' &= b_2 z_1 + b_1 z_2 + b_0 z_3,\\ z_2' &= z_1\\ z_3' &= z_2, \end{aligned}

or

\notag \begin{bmatrix} z_1 \\ z_2\\ z_3 \end{bmatrix}' = \begin{bmatrix} b_2 & b_1 & b_0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{bmatrix} \begin{bmatrix} z_1 \\ z_2\\ z_3 \end{bmatrix},

so the third order scalar equation has been converted into a first order system with a companion matrix as coefficient matrix.

Computing Polynomial Roots

The MATLAB function roots takes as input a vector of the coefficients of a polynomial and returns the roots of the polynomial. It computes the eigenvalues of the companion matrix associated with the polynomial using the eig function. As Moler (1991) explained, MATLAB used this approach starting from the first version of MATLAB, but it does not take advantage of the structure of the companion matrix, requiring O(n^3) flops and O(n^2) storage instead of the O(n^2) flops and O(n) storage that should be possible given the structure of C. Since the early 2000s much research has aimed at deriving methods that achieve this objective, but numerically stable methods proved elusive. Finally, a backward stable algorithm requiring O(n^2) flops and O(n) storage was developed by Aurentz, Mach, Vandebril, and Watkins (2015). It uses the QR algorithm and exploits the unitary plus low rank structure shown in (1). Here, backward stability means that the computed roots are the eigenvalues of C + \Delta C for some \Delta C with \|\Delta C\| \le c_n u \|C\|. It is not necessarily the case that the computed roots are the exact roots of a polynomial with coefficients a_i + \Delta a_i with |\Delta a_i| \le c_n u \max_i |a_i| for all i.

Rational Canonical Form

It is an interesting observation that

\notag \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & -a_3\\ 0 & 1 & -a_3 & -a_2 \\ 1 & -a_3 & -a_2 & -a_1 \end{bmatrix} \begin{bmatrix} a_3 & a_2 & a_1 & a_0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & -a_3 & 0 \\ 1 & -a_3 & -a_2 & 0 \\ 0 & 0 & 0 & a_0 \\ \end{bmatrix}.

Multiplying by the inverse of the matrix on the left we express the 4\times 4 companion matrix as the product of two symmetric matrices. The obvious generalization of this factorization to n\times n matrices shows that we can write

\notag C = S_1S_2, \quad S_1 = S_1^T, \quad S_2= S_2^T. \qquad (2)

We need the rational canonical form of a matrix, described in the next theorem, which Halmos (1991) calls “the deepest theorem of linear algebra”. Let \mathbb{F} denote the field \mathbb{R} or \mathbb{C}.

Theorem 1 (rational canonical form).

If A\in\mathbb{F}^{n\times n} then A = X^{-1} C X where X\in\mathbb{F}^{n\times n} is nonsingular and C = \mathrm{diag}(C_i)\in\mathbb{F}^{n\times n}, with each C_i a companion matrix.

The theorem says that every matrix is similar over the underlying field to a block diagonal matrix composed of companion matrices. Since we do not need it, we have omitted from the statement of the theorem the description of the C_i in terms of the irreducible factors of the characteristic polynomial. Combining the factorization (2) and Theorem 1 we obtain

\notag \begin{aligned} A &= X^{-1}CX = X^{-1} S_1S_2 X \\ & = X^{-1}S_1X^{-T} \cdot X^T S_2 X \\ & \equiv \widetilde{S}_1 \widetilde{S}_2,\quad \widetilde{S}_1 = \widetilde{S}_1^T, \quad \widetilde{S}_2 = \widetilde{S}_2^T. \end{aligned}

Since S_1 is nonsingular, and since S_2 can alternatively be taken nonsingular by considering the factorization of A^T, this proves a theorem of Frobenius.

Theorem 2 (Frobenius, 1910).

For any A\in\mathbb{F}^{n\times n} there exist symmetric S_1,S_2\in\mathbb{F}^{n\times n}, either one of which can be taken nonsingular, such that A = S_1 S_2.

Note that if A = S_1S_2 with the S_i symmetric then AS_1 = S_1S_2S_1 = S_1A^T = (AS_1)^T, so AS_1 is symmetric. Likewise, S_2A is symmetric.

Factorization

Fiedler (2003) noted that a companion matrix can be factorized into the product of n simpler factors, n-1 of them being the identity matrix with a 2\times 2 block placed on the diagonal, and he used this factorization to determine a matrix \widetilde{C} similar to C. For n = 5 it is

\notag \widetilde{C} = \begin{bmatrix} a_4 & a_3 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & a_2 & 0 & a_1 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & a_0 & 0 \end{bmatrix} = \left[\begin{array}{cc|cc|c} a_4 & 1 & & & \\ 1 & 0 & & & \\\hline & & a_2 & 1 & \\ & & 1 & 0 & \\\hline & & & & a_0 \end{array}\right] \left[\begin{array}{c|cc|cc} 1 & & & & \\\hline & a_3 & 1 & & \\ & 1 & 0 & & \\\hline & & & a_1 & 1 \\ & & & 1 & 0 \end{array}\right].

In general, Fielder’s construction yields an n\times n pentadiagonal matrix \widetilde{C} that is not simply a permutation similarity of C. The fact that \widetilde{C} has block diagonal factors opens the possibility of obtaining new methods for finding the eigenvalues of C. This line of research has been extensively pursued in the context of polynomial eigenvalue problems (see Mackey, 2013).

Generalizations

The companion matrix is associated with the monomial basis representation of the characteristic polynomial. Other polynomial bases can be used, notably orthogonal polynomials, and this leads to generalizations of the companion matrix having coefficients on the main diagonal and the subdiagonal and superdiagonal. Good (1961) calls the matrix resulting from the Chebyshev basis a colleague matrix. Barnett (1981) calls the matrices corresponding to orthogonal polynomials comrade matrices, and for a general polynomial basis he uses the term confederate matrices. Generalizations of the properties of companion matrices can be derived for these classes of matrices.

Bounds for Polynomial Roots

Since the roots of a polynomial are the eigenvalues of the associated companion matrix, or a Fiedler matrix similar to it, or indeed the associated comrade matrix or confederate matrix, one can obtain bounds on the roots by applying any available bounds for matrix eigenvalues. For example, since any eigenvalue \lambda of matrix A satisfies |\lambda| \le \|A\|, by taking the 1-norm and the \infty-norm of the companion matrix C we find that any root \lambda of the polynomial (*) satisfies

\notag \begin{aligned} |\lambda| &\le \max\bigl(|a_0|, 1 + \max_{j = 1:n-1} |a_j| \bigr), \\ |\lambda| &\le \max(1, |a_{n-1}| + |a_{n-2}| + \cdots + |a_0|), \end{aligned}

either of which can be the smaller. A rich variety of such bounds is available, and these techniques extend to matrix polynomials and the corresponding block companion matrices.

References

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

Related Blog Posts

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

What Is an M-Matrix?

An M-matrix is a matrix A\in\mathbb{R}^{n \times n} of the form

\notag A = sI - B, \quad \mathrm{where}~B \ge 0~\mathrm{and}~s > \rho(B). \qquad (*)

Here, \rho(B) is the spectral radius of B, that is, the largest modulus of any eigenvalue of B, and B \ge 0 denotes that B has nonnegative entries. An M-matrix clearly has nonpositive off-diagonal elements. It also has positive diagonal elements, which can be shown using the result that

\notag \rho(A) = \lim_{k\to\infty} \|A^k\|^{1/k} \qquad (\dagger)

for any consistent matrix norm:

\notag s > \rho(B) = \lim_{k\to\infty} \|B^k\|_{\infty}^{1/k} \ge \lim_{k\to\infty} \|\mathrm{diag}(b_{ii})^k\|_{\infty}^{1/k} = \max_i b_{ii}.

Although the definition of an M-matrix does not specify s, we can set it to \max_i a_{ii}. Indeed let s satisfy (*) and set \widetilde{s} = \max_i a_{ii} and \widetilde{B} = \widetilde{s}I - A. Then \widetilde{B} \ge 0, since \widetilde{b}_{ii} = \widetilde{s} - a_{ii} \ge 0 and \widetilde{b}_{ij} = -a_{ij} = b_{ij} \ge 0 for i \ne j. Furthermore, for a nonnegative matrix the spectral radius is an eigenvalue, by the Perron–Frobenius theorem, so \rho(B) is an eigenvalue of B and \rho(\widetilde{B)} is an eigenvalue of \widetilde{B}. Hence \rho(\widetilde{B}) = \rho( (\widetilde{s}-s)I + B) = \widetilde{s} -s + \rho(B) < \widetilde{s}.

The concept of M-matrix was introduced by Ostrowski in 1937. M-matrices arise in a variety of scientific settings, including in finite difference methods for PDEs, input-output analysis in economics, and Markov chains in stochastic processes.

An immediate consequence of the definition is that the eigenvalues of an M-matrix lie in the open right-half plane, which means that M-matrices are special cases of positive stable matrices. Hence an M-matrix is nonsingular and the determinant, being the product of the eigenvalues, is positive. Moreover, since C = s^{-1}B satisfies \rho(C) < 1,

\notag A^{-1} = s^{-1}(I - C)^{-1} = s^{-1}(I + C + C^2 + \cdots) \ge 0.

In fact, nonnegativity of the inverse characterizes M-matrices. Define

\notag Z_n = \{ \, A \in \mathbb{R}^{n\times n}: a_{ij} \le 0, \; i\ne j \,\}.

Theorem 1.

A nonsingular matrix A\in Z_n is an M-matrix if and only if A^{-1} \ge 0.

Sometimes an M-matrix is defined to be a matrix with nonpositive off-diagonal elements and a nonnegative inverse. In fact, this condition is just one of a large number of conditions equivalent to a matrix with nonpositive off-diagonal elements being an M-matrix, fifty of which are given in Berman and Plemmons (1994, Chap. 6).

It is easy to check from the definitions, or using Theorem 1, that a triangular matrix T with positive diagonal and nonpositive off-diagonal is an M-matrix. An example is

\notag T_4 = \left[\begin{array}{@{\mskip 5mu}c*{3}{@{\mskip 15mu} r}@{\mskip 5mu}} 1 & -1 & -1 & -1 \\ & 1 & -1 & -1 \\ & & 1 & -1 \\ & & & 1 \end{array}\right], \quad T_4^{-1} = \begin{bmatrix} 1 & 1 & 2 & 4\\ & 1 & 1 & 2\\ & & 1 & 1\\ & & & 1 \end{bmatrix}.

Bounding the Norm of the Inverse

An M-matrix can be constructed from any nonsingular triangular matrix by taking the comparison matrix. The comparison matrix associated with a general B\in\mathbb{R}^{n \times n} is the matrix

\notag M(B) = (m_{ij}), \quad m_{ij} = \begin{cases} |b_{ii}|, & i=j, \\ -|b_{ij}|, & i\ne j. \end{cases}

For a nonsingular triangular T, M(T) is an M-matrix, and it easy to show that

\notag |T^{-1}| \le |M(T)^{-1}|,

where the absolute value is taken componentwise. This bound, and weaker related bounds, can be useful for cheaply bounding the norm of the inverse of a triangular matrix. For example, with e denoting the vector of ones, since M(T)^{-1} is nonnegative we have

\notag \|T^{-1}\|_{\infty} \le \|M(T)^{-1}\|_{\infty} = \|M(T)^{-1}e\|_{\infty},

and \|M(T)^{-1}e\|_{\infty} can be computed in O(n^2) flops by solving a triangular system, whereas computing T^{-1} costs O(n^3) flops.

More generally, if we have an LU factorization PA = LU of an M-matrix A\in\mathbb{R}^{n \times n} then, since A^{-1} \ge 0,

\notag \|A^{-1}\|_{\infty} = \|A^{-1}e\|_{\infty} = \|U^{-1}L^{-1}Pe\|_{\infty} = \|U^{-1}L^{-1}e\|_{\infty}.

Therefore the norm of the inverse can be computed in O(n^2) flops with two triangular solves, instead of the O(n^3) flops that would be required if A^{-1} were to be formed explicitly.

Connections with Symmetric Positive Definite Matrices

There are many analogies between M-matrices and symmetric positive definite matrices. For example, every principal submatrix of a symmetric positive definite matrix is symmetric positive definite and every principal submatrix of an M-matrix is an M-matrix. Indeed if \widetilde{B} is a principal submatrix of a nonnegative B then \rho(\widetilde{B}) \le \rho(B), which follows from (\dagger) for the \infty-norm (for example). Hence on taking principal submatrices in (*) we have s > \rho(\widetilde{B}) with the same s.

A symmetric M-matrix is known as a Stieltjes matrix, and such a matrix is positive definite. An example of a Stieltjes matrix is minus the second difference matrix (the tridiagonal matrix arising from a central difference discretization of a second derivative), illustrated for n = 4 by

\notag A_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], \quad A_4^{-1} = \begin{bmatrix} \frac{4}{5} & \frac{3}{5} & \frac{2}{5} & \frac{1}{5}\\[\smallskipamount] \frac{3}{5} & \frac{6}{5} & \frac{4}{5} & \frac{2}{5}\\[\smallskipamount] \frac{2}{5} & \frac{4}{5} & \frac{6}{5} & \frac{3}{5}\\[\smallskipamount] \frac{1}{5} & \frac{2}{5} & \frac{3}{5} & \frac{4}{5} \end{bmatrix}.

LU Factorization

Since the leading principal submatrices of an M-matrix A have positive determinant it follows that A has an LU factorization with U having positive diagonal elements. However, more is true, as the next result shows.

Theorem 2.

An M-matrix A has an LU factorization A = LU in which L and U are M-matrices.

Proof. We can write

\notag A = \begin{array}[b]{@{\mskip27mu}c@{\mskip-20mu}c@{\mskip-10mu}c@{}} \scriptstyle 1 & \scriptstyle n-1 & \\ \multicolumn{2}{c}{ \left[\begin{array}{c@{~}c@{~}} \alpha & b^T \\ c^T & E \\ \end{array}\right]} & \mskip-14mu\ \begin{array}{c} \scriptstyle 1 \\ \scriptstyle n-1 \end{array} \end{array}, \quad \alpha > 0, \quad b\le 0, \quad c \le 0.

The first stage of LU factorization is

\notag A = \begin{bmatrix} \alpha & b^T \\ c & E \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ \alpha^{-1}c & I \end{bmatrix} \begin{bmatrix} \alpha & b^T \\ 0 & S \end{bmatrix} = L_1U_1, \quad S = E - \alpha^{-1} c\mskip1mu b^T,

where S is the Schur complement of \alpha in A. The first column of L_1 and the first row of U_1 are of the form required for a triangular M-matrix. We have

\notag A^{-1} = U_1^{-1}L_1^{-1} = \begin{bmatrix} \alpha^{-1} & -\alpha^{-1}b^TS^{-1} \\ 0 & S^{-1} \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -\alpha^{-1}c & I \end{bmatrix} = \begin{bmatrix} \times & \times \\ \times & S^{-1} \end{bmatrix}.

Since A^{-1} \ge 0 it follows that S^{-1} \ge 0. It is easy to see that S\in Z_n, and hence Theorem 1 shows that S is an M-matrix. The result follows by induction.

The question now arises of what can be said about the numerical stability of LU factorization of an M-matrix. To answer it we use another characterization of M-matrices, that DA is strictly diagonally dominant by columns for some diagonal matrix D = \mathrm{diag}(d_i) with d_i>0 for all i, that is,

\notag d_j|a_{jj}| > \sum_{i\ne j} d_i |a_{ij}|, \quad j=1\colon n.

(This condition can also be written as d^TA > 0 because of the sign pattern of A.) Matrices that are diagonally dominant by columns have the properties that an LU factorization without pivoting exists, the growth factor \rho_n \le 2, and partial pivoting does not require row interchanges. The effect of row scaling on LU factorization is easy to see:

\notag A = LU \;\Rightarrow\; DA = DLD^{-1} \cdot DU \equiv \widetilde{L} \widetilde{U},

where \widetilde{L} is unit lower triangular, so that \widetilde{L} and \widetilde{U} are the LU factors of DA. It is easy to see that the growth factor bound of 2 for a matrix diagonally dominant by columns translates into the bound

\notag \rho_n \le 2\mskip1mu\displaystyle\frac{\max_i d_i}{\min_i d_i} \qquad(\ddagger)

for an M-matrix, which was obtained by Funderlic, Neumann, and Plemmons (1982). Unfortunately, this bound can be large. Consider the matrix

\notag A = \begin{bmatrix} \epsilon& 0& -1\\ -1& 1& -1\\ 0& 0& 1 \end{bmatrix} \in Z_3, \quad 0 < \epsilon < 1.

We have

\notag A^{-1} = \begin{bmatrix} \displaystyle\frac{1}{\epsilon}& 0& \displaystyle\frac{1}{\epsilon}\\[\bigskipamount] \displaystyle\frac{1}{\epsilon}& 1 & \displaystyle\frac{1 + \epsilon}{\epsilon}\\[\bigskipamount] 0& 0& 1 \end{bmatrix} \ge 0,

so A is an M-matrix. The (2,3) element of the LU factor U of A is -1 - 1/\epsilon, which means that

\notag \rho_3 \ge \displaystyle\frac{1}{\epsilon} + 1,

and this lower bound can be arbitrarily large. One can verify experimentally that numerical instability is possible when \rho_3 is large, in that the computed LU factors have a large relative residual. We conclude that pivoting is necessary for numerical stability in LU factorization of M-matrices.

Stationery Iterative Methods

A stationery iterative method for solving a linear system Ax = b is based on a splitting A = M - N with M nonsingular, and has the form Mx_{k+1} = Nx_k + b. This iteration converges for all starting vectors x_0 if \rho(M^{-1}N) < 1. Much interest has focused on regular splittings, which are defined as ones for which M^{-1}\ge 0 and N \ge 0. An M-matrix has the important property that \rho(M^{-1}N) < 1 for every regular splitting, and it follows that the Jacobi iteration, the Gauss-Seidel iteration, and the successive overrelaxation (SOR) iteration (with 0 < \omega \le 1) are all convergent for M-matrices.

Matrix Square Root

The principal square root A^{1/2} of an M-matrix A is an M-matrix, and it is the unique such square root. An expression for A^{1/2} follows from (*):

\notag \begin{aligned} A^{1/2} &= s^{1/2}(I - C)^{1/2} \quad (C = s^{-1}B, ~\rho(C) < 1),\\ &= s^{1/2} \sum_{j=0}^{\infty} {\frac{1}{2} \choose j} (-C)^j. \end{aligned}

This expression does not necessarily provide the best way to compute A^{1/2}.

Singular M-Matrices

The theory of M-matrices extends to the case where the condition on s is relaxed to s \ge \rho(B) in (*), though the theory is more complicated and extra conditions such as irreducibility are needed for some results. Singular M-matrices occur in Markov chains (Berman and Plemmons, 1994, Chapter 8), for example. Let P be the transition matrix of a Markov chain. Then P is stochastic, that is, nonnegative with unit row sums, so Pe = e. A nonnegative vector y with y^Te = 1 such that y^T P = y^T is called a stationary distribution vector and is of interest for describing the properties of the Markov chain. To compute y we can solve the singular system Ay = (I - P^T)y = 0. Clearly, A\in Z_n and \rho(P) = 1, so A is a singular M-matrix.

H-Matrices

A more general concept is that of H-matrix: A\in\mathbb{R}^{n \times n} is an H-matrix if the comparison matrix M(A) is an M-matrix. Many results for M-matrices extend to H-matrices. For example, for an H-matrix with positive diagonal elements the principal square root exists and is the unique square root that is an H-matrix with positive diagonal elements. Also, the growth factor bound (\ddagger) holds for any H-matrix for which DA is diagonally dominant by columns.

References

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

Related Blog Posts

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

Eigenvalue Inequalities for Hermitian Matrices

The eigenvalues of Hermitian matrices satisfy a wide variety of inequalities. We present some of the most useful and explain their implications. Proofs are omitted, but as Parlett (1998) notes, the proofs of the Courant–Fischer, Weyl, and Cauchy results are all consequences of the elementary fact that if the sum of the dimensions of two subspaces of \mathbb{C}^n exceeds n then the subspaces have a nontrivial intersection.

The eigenvalues of a Hermitian matrix A\in\mathbb{C}^{n\times n} are real and we order them \lambda_n\le \lambda_{n-1} \le \cdots \le \lambda_1. Note that in some references, such as Horn and Johnson (2013), the reverse ordering is used, with \lambda_n the largest eigenvalue. When it is necessary to specify what matrix \lambda_k is an eigenvalue of we write \lambda_k(A): the kth largest eigenvalue of A. All the following results also hold for symmetric matrices over \mathbb{R}^{n\times n}.

Quadratic Form

The function f(x) = x^*Ax/x^*x is the quadratic form x^*Ax for A evaluated on the unit sphere, since f(x) = f(x/\|x\|_2). As A is Hermitian it has a spectral decomposition A = Q\Lambda Q^*, where Q is unitary and \Lambda = \mathrm{diag}(\lambda_i). Then

f(x) = \displaystyle\frac{x^*Q\Lambda Q^*x}{x^*x} = \displaystyle\frac{y^*\Lambda y}{y^*y} = \displaystyle\frac{\sum_{i=1}^{n}\lambda_i y_i^2} {\sum_{i=1}^{n}y_i^2} \quad (y = Q^*x),

from which is it clear that

\notag \lambda_n = \displaystyle\min_{x\ne0} \displaystyle\frac{x^*Ax}{x^*x}, \quad \lambda_1 = \displaystyle\max_{x\ne0} \displaystyle\frac{x^*Ax}{x^*x}, \qquad(*)

with equality when x is an eigenvector corresponding to \lambda_n and \lambda_1, respectively, This characterization of the extremal eigenvalues of A as the extrema of f is due to Lord Rayleigh (John William Strutt), and f(x) is called a Rayleigh quotient. The intermediate eigenvalues correspond to saddle points of f.

Courant–Fischer Theorem

The Courant–Fischer theorem (1905) states that every eigenvalue of a Hermitian matrix A\in\mathbb{C}^{n\times n} is the solution of both a min-max problem and a max-min problem over suitable subspaces of \mathbb{C}^n.

Theorem (Courant–Fischer).

For a Hermitian A\in\mathbb{C}^{n\times n},

\notag \begin{aligned} \lambda_k &= \min_{\dim(S)=n-k+1} \, \max_{0\ne x\in S} \frac{x^*Ax}{x^*x}\\ &= \max_{\dim(S)= k} \, \min_{0\ne x\in S} \frac{x^*Ax}{x^*x}, \quad k=1\colon n. \end{aligned}

Note that the equalities (*) are special cases of these characterizations.

In general there is no useful formula for the eigenvalues of a sum A+B of Hermitian matrices. However, the Courant–Fischer theorem yields the upper and lower bounds

\notag \lambda_k(A) + \lambda_n(B) \le \lambda_k(A+B) \le \lambda_k(A) + \lambda_1(B), \qquad (1)

from which it follows that

\notag \max_k|\lambda_k(A+B)-\lambda_k(A)| \le \max(|\lambda_n(B)|,|\lambda_1(B)|) = \|B\|_2.

This inequality shows that the eigenvalues of a Hermitian matrix are well conditioned under perturbation. We can rewrite the inequality in the symmetric form

\notag \max_k |\lambda_k(A)-\lambda_k(B)| \le \|A-B\|_2.

If B is positive semidefinite then (1) gives

\notag \lambda_k(A) \le \lambda_k(A + B), \quad k = 1\colon n, \qquad (2)

while if B is positive definite then strict inequality holds for all i. These bounds are known as the Weyl monotonicity theorem.

Weyl’s Inequalities

Weyl’s inequalities (1912) bound the eigenvalues of A+B in terms of those of A and B.

Theorem (Weyl).

For Hermitian A,B\in\mathbb{C}^{n\times n} and i,j = 1\colon n,

\notag \begin{aligned} \lambda_{i+j-1}(A+B) &\le \lambda_i(A) + \lambda_j(B), \quad i+j \le n+1, \qquad (3)\\ \lambda_i(A) + \lambda_j(B) &\le \lambda_{i+j-n}(A+B). \quad i+j \ge n+1, \qquad (4) \end{aligned}

The Weyl inequalities yield much information about the effect of low rank perturbations. Consider a positive semidefinite rank-1 perturbation B = zz^*. Inequality (3) with j = 1 gives

\notag \lambda_i(A+B) \le \lambda_i(A) + z^*z, \quad i = 1\colon n

(which also follows from (1)). Inequality (3) with j = 2, combined with (2), gives

\notag \lambda_{i+1}(A) \le \lambda_{i+1}(A + zz^*) \le \lambda_i(A), \quad i = 1\colon n-1. \qquad (5)

These inequalities confine each eigenvalue of A + zz^* to the interval between two adjacent eigenvalues of A; the eigenvalues of A + zz^* are said to interlace those of A. The following figure illustrates the case n = 4, showing a possible configuration of the eigenvalues \lambda_i of A and \mu_i of A + zz^*.

weyl_fig.jpg A specific example, in MATLAB, is

>> n = 4; eig_orig = 5:5+n-1
>> D = diag(eig_orig); eig_pert = eig(D + ones(n))'
eig_orig =
     5     6     7     8
eig_pert =
   5.2961e+00   6.3923e+00   7.5077e+00   1.0804e+01

Since \mathrm{trace}(A + zz^*) = \mathrm{trace}(A) + z^*z and the trace is the sum of the eigenvalues, we can write

\notag \lambda_i(A + zz^*) = \lambda_i(A) + \theta_i z^*z,

where the \theta_i are nonnegative and sum to 1. If we greatly increase z^*z, the norm of the perturbation, then most of the increase in the eigenvalues is concentrated in the largest, since (5) bounds how much the smaller eigenvalues can change:

>> eig_pert = eig(D + 100*ones(n))'
eig_pert =
   5.3810e+00   6.4989e+00   7.6170e+00   4.0650e+02

More generally, if B has p positive eigenvalues and q negative eigenvalues then (3) with j = p+1 gives

\notag \lambda_{i+p}(A+B) \le \lambda_i(A), \quad i = 1\colon n-p,

while (4) with j = n-q gives

\notag \lambda_i(A) \le \lambda_{i-q}(A + B), \quad i = q+1\colon n.

So the inertia of B (the number of negative, zero, and positive eigenvalues) determines how far the eigenvalues can move as measured relative to the indexes of the eigenvalues of A.

An important implication of the last two inequalities is for the case A = I, for which we have

\notag \begin{aligned} \lambda_{i+p}(I+B) &\le 1, \quad i = 1 \colon n-p, \\ \lambda_{i-q}(I+B) &\ge 1, \quad i = q+1 \colon n. \end{aligned}

Exactly p+q eigenvalues appear in one of these inequalities and n-(p+q) appear in both. Therefore n - (p+q) of the eigenvalues are equal to 1 and so only \mathrm{rank}(B) = p+q eigenvalues can differ from 1. So perturbing the identity matrix by a Hermitian matrix of rank r changes at most r of the eigenvalues. (In fact, it changes exactly r eigenvalues, as can be seen from a spectral decomposition.)

Finally, if B has rank r then \lambda_{r+1}(B) \le 0 and \lambda_{n-r}(B) \ge 0 and so taking j = r+1 in (3) and j = n-r in (4) gives

\notag \begin{aligned} \lambda_{i+r}(A+B) &\le \lambda_i(A), ~~\qquad\qquad i = 1\colon n-r, \\ \lambda_i(A) &\le \lambda_{i-r}(A + B), ~~\quad i = r+1\colon n. \end{aligned}

Cauchy Interlace Theorem

The Cauchy interlace theorem relates the eigenvalues of successive leading principal submatrices of a Hermitian matrix. We denote the leading principal submatrix of A of order k by A_k = A(1\colon k, 1\colon k).

Theorem (Cauchy).

For a Hermitian A\in\mathbb{C}^{n\times n},

\notag \lambda_{i+1}(A_{k+1}) \le \lambda_i(A_k) \le \lambda_i(A_{k+1}), \quad i = 1\colon k, \quad k=1\colon n-1.

The theorem says that the eigenvalues of A_k interlace those of A_{k+1} for all k. Two immediate implications are that (a) if A is Hermitian positive definite then so are all its leading principal submatrices and (b) appending a row and a column to a Hermitian matrix does not decrease the largest eigenvalue or increase the smallest eigenvalue.

Since eigenvalues are unchanged under symmetric permutations of the matrix, the theorem can be reformulated to say that the eigenvalues of any principal submatrix of order n-1 interlace those of A. A generalization to principal submatrices of order n-\ell is given in the next result.

Theorem.

If B is a principal submatrix of order n-\ell of a Hermitian A\in\mathbb{C}^{n\times n} then

\notag \lambda_{i+\ell}(A) \le \lambda_i(B) \le \lambda_i(A), \quad i=1\colon n-\ell.

Majorization Results

It follows by taking x to be a unit vector e_i in the formula \lambda_1 = \max_{x\ne0} x^*Ax/(x^*x) that \lambda_1 \ge a_{ii} for all i. And of course the trace of A is the sum of the eigenvalues: \sum_{i=1}^n a_{ii} = \sum_{i=1}^n \lambda_i. These relations are the first and last in a sequence of inequalities relating sums of eigenvalues to sums of diagonal elements obtained by Schur in 1923.

Theorem (Schur).

For a Hermitian A\in\mathbb{C}^{n\times n},

\notag \displaystyle\sum_{i=1}^k \lambda_i \ge \displaystyle\sum_{i=1}^k \widetilde{a}_{ii}, \quad k=1\colon n,

where \{\widetilde{a}_{ii}\} is the set of diagonal elements of A arranged in decreasing order: \widetilde{a}_{11} \ge \cdots \ge \widetilde{a}_{nn}.

These inequalities say that the vector [\lambda_1,\dots,\lambda_n] of eigenvalues majorizes the ordered vector [\widetilde{a}_{11},\dots,\widetilde{a}_{nn}] of diagonal elements.

An interesting special case is a correlation matrix, a symmetric positive semidefinite matrix with unit diagonal, for which the inequalities are

\notag \lambda_1 \ge 1, \quad \lambda_1+ \lambda_2\ge 2, \quad \dots, \quad \lambda_1+ \lambda_2 + \cdots + \lambda_{n-1} \ge n-1,

and \lambda_1+ \lambda_2 + \cdots + \lambda_n = n. Here is an illustration in MATLAB.

>> n = 5; rng(1); A = gallery('randcorr',n);
>> e = sort(eig(A)','descend'), partial_sums = cumsum(e)
e =
  2.2701e+00   1.3142e+00   9.5280e-01   4.6250e-01   3.6045e-04
partial_sums =
  2.2701e+00   3.5843e+00   4.5371e+00   4.9996e+00   5.0000e+00

Ky Fan (1949) proved a majorization relation between the eigenvalues of A, B, and A+B:

\notag \displaystyle\sum_{i=1}^k \lambda_i(A+B) \le \displaystyle\sum_{i=1}^k \lambda_i(A) + \displaystyle\sum_{i=1}^k \lambda_i(B), \quad k = 1\colon n.

For k = 1, the inequality is the same as the upper bound of (1), and for k = n it is an equality: \mathrm{trace}(A+B) = \mathrm{trace}(A) + \mathrm{trace}(B).

Ostrowski’s Theorem

For a Hermitian A and a nonsingular X, the transformation A\to X^*AX is a congruence transformation. Sylvester’s law of inertia says that congruence transformations preserve the inertia. A result of Ostrowski (1959) goes further by providing bounds on the ratios of the eigenvalues of the original and transformed matrices.

Theorem (Ostrowski).

For a Hermitian A\in \mathbb{C}^{n\times n} and X\in\mathbb{C}^{n\times n},

\lambda_k(X^*AX) = \theta_k \lambda_k(A), \quad k=1\colon n,

where \lambda_n(X^*X) \le \theta_k \le \lambda_1(X^*X).

If X is unitary then X^*X = I and so Ostrowski’s theorem reduces to the fact that a congruence with a unitary matrix is a similarity transformation and so preserves eigenvalues. The theorem shows that the further X is from being unitary the greater the potential change in the eigenvalues.

Ostrowski’s theorem can be generalized to the situation where X is rectangular (Higham and Cheng, 1998).

Interrelations

The results we have described are strongly interrelated. For example, the Courant–Fischer theorem and the Cauchy interlacing theorem can be derived from each other, and Ostrowski’s theorem can be proved using the Courant–Fischer Theorem.

References

SIAM CSE21 Minisymposium on Reduced Precision Arithmetic and Stochastic Rounding

This minisymposium took place at the SIAM Conference on Computational Science and Engineering, March 2, 2021. This page makes available slides from some of the talks.

Minisymposium description: Reduced precision floating-point arithmetic, such as IEEE half precision and bfloat16, is increasingly available in hardware. Low precision computations promise major increases in speed and reductions in data communication costs, but they also bring an increased risk of overflow, underflow, and loss of accuracy. One way to improve the results of low precision computations is to use stochastic rounding instead of round to nearest, and this is proving popular in machine learning. This minisymposium will discuss recent advances in exploitation and analysis of reduced precision arithmetic and stochastic rounding.

Algorithms for Stochastically Rounded Elementary Arithmetic Operations in IEEE 754 Floating-Point Arithmetic Massimiliano Fasi, Örebro University, Sweden; Mantas Mikaitis, University of Manchester, United Kingdom. Abstract. Slides.

Reduced Precision Elementary Functions Jean-Michel Muller, ENS Lyon, France. Abstract. Slides.

Effect of Reduced Precision and Stochastic Rounding in the Numerical Solution of Parabolic Equations Matteo Croci and Michael B. Giles, University of Oxford, United Kingdom. Abstract. Slides.

Stochastic Rounding and its Probabilistic Backward Error Analysis Michael P. Connolly and Nicholas J. Higham, University of Manchester, United Kingdom; Theo Mary, Sorbonne Universités and CNRS, France. Abstract. Slides.

Stochastic Rounding in Weather and Climate Models Milan Kloewer, Edmund Paxton, and Matthew Chantry, University of Oxford, United Kingdom Abstract. Slides.