On approximately simultaneously diagonalizable matrices

A collection A₁, A₂, …, A_k of n × n matrices over the complex numbers C has the ASD property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable. Such a collection must perforce be commuting. We show by a direct matrix proof that the ASD property holds for three commuting matrices when one of them is 2-regular (dimension of eigenspaces is at most 2). Corollaries include results of Gerstenhaber and Neubauer-Sethuraman on bounds for the dimension of the algebra generated by A₁, A₂, …, A_k. Even when the ASD property fails, our techniques can produce a good bound on the dimension of this subalgebra. For example, we establish for commuting matrices A₁, …, A_k when one of them is 2-regular. This bound is sharp. One offshoot of our work is the introduction of a new canonical form, the H-form, for matrices over an algebraically closed field. The H-form of a matrix is a sparse “Jordan like” upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular. (The Jordan form itself does not accommodate this.)     

Fine spectra of upper triangular double-band matrices

The fine spectra of lower triangular double-band matrices have been examined by several authors (e.g. [13] and [22]). Here we determine the fine spectra of upper triangular double-band matrices over the sequence spaces c₀ and c. Upper triangular double-band matrices are infinite matrices which include the left-shift, averaging and difference operators.     

Infinite triangular matrices, q-Pascal matrices, and determinantal representations

We use basic properties of infinite lower triangular matrices and the connections of Toeplitz matrices with generating-functions to obtain inversion formulas for several types of q-Pascal matrices, determinantal representations for polynomial sequences, and identities involving the q-Gaussian coefficients. We also obtain a fast inversion algorithm for general infinite lower triangular matrices.     

Subtotally positive and Monge matrices

A real matrix is called k-subtotally positive if the determinants of all its submatrices of order at most k are positive. We show that for an m × n matrix, only mn inequalities determine such class for every k, 1 ? k ? min(m,n). Spectral properties of square k-subtotally positive matrices are studied. Finally, completion problems for 2-subtotally positive matrices and their additive counterpart, the anti-Monge matrices, are investigated. Since totally positive matrices are 2-subtotally positive as well, the presented necessary conditions for this completion problem are also necessary conditions for totally positive matrices.     

A criterion for unitary similarity of upper triangular matrices in general position

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A=[a_ij] and B=[b_ij] be upper triangular n×n matrices that

•

are not similar to direct sums of square matrices of smaller sizes, or

•

are in general position and have the same main diagonal.

We prove that A and B are unitarily similar if and only if

     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

On extremal matrices of second largest exponent by Boolean rank

Let b = b(A) be the Boolean rank of an n × n primitive Boolean matrix A and exp(A) be the exponent of A. Then exp(A) ? (b − 1)² + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101-116]. In this paper, we show that for each 3 ? b ? n − 1, there are n × n primitive Boolean matrices A with b(A) = b such that exp(A) = (b − 1)² + 1, and we explicitly describe all such matrices.     

A note on the perturbation of positive matrices by normal and unitary matrices

In a recent paper, Neumann and Sze considered for an n × n nonnegative matrix A, the minimization and maximization of ρ(A + S), the spectral radius of (A + S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n × n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A + N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A + N) as N ranges over all such matrices. We show that the extremal values always occur at an n × n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A + X) over all n × n real matrices X of Frobenius norm was sought.     

Schur product of matrices and numerical radius (range) preserving maps

Let F(A) be the numerical range or the numerical radius of a square matrix A. Denote by A ° B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F(A ° B) = F(?(A) ° ?(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices.     

Factorization of singular integer matrices

It is well known that a singular integer matrix can be factorized into a product of integer idempotent matrices. In this paper, we prove that every n  × n (n > 2) singular integer matrix can be written as a product of 3n + 1 integer idempotent matrices. This theorem has some application in the field of synthesizing VLSI arrays and systolic arrays.     

An asymptotic behavior of QR decomposition

The m-th root of the diagonal of the upper triangular matrix R_m in the QR decomposition of AX^mB = Q_mR_m converges and the limit is given by the moduli of the eigenvalues of X with some ordering, where A,B,X∈C_n×n are nonsingular. The asymptotic behavior of the strictly upper triangular part of R_m is discussed. Some computational experiments are discussed.     

The eigenvalue distribution on Schur complements of H-matrices

The paper studies the eigenvalue distribution of some special matrices. Tong in Theorem 1.2 of [Wen-ting Tong, On the distribution of eigenvalues of some matrices, Acta Math. Sinica (China), 20 (4) (1977) 273-275] gives conditions for an n × n matrix A ∈ SD_n ∪ ID_n to have |J_R+(A)| eigenvalues with positive real part, and |J_R-(A)| eigenvalues with negative real part. A counter-example is given in this paper to show that the conditions of the theorem are not true. A corrected condition is then proposed under which the conclusion of the theorem holds. Then the corrected condition is applied to establish some results about the eigenvalue distribution of the Schur complements of H-matrices with complex diagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, … , n} are presented such that the Schur complement matrix A/α of the matrix A has eigenvalues with positive real part and eigenvalues with negative real part.     

On block partial linearizations of the pfaffian

Amitsur’s formula, which expresses det(A + B) as a polynomial in coefficients of the characteristic polynomial of a matrix, is generalized for partial linearizations of the pfaffian of block matrices. As applications, in upcoming papers we determine generators for the SO(n)-invariants of several matrices and relations for the O(n)-invariants of several matrices over a field of arbitrary characteristic.     

Multiple LU factorizations of a singular matrix

A singular matrix A may have more than one LU factorizations. In this work the set of all LU factorizations of A is explicitly described when the lower triangular matrix L is nonsingular. To this purpose, a canonical form of A under left multiplication by unit lower triangular matrices is introduced. This canonical form allows us to characterize the matrices that have an LU factorization and to parametrize all possible LU factorizations. Formulae in terms of quotient of minors of A are presented for the entries of this canonical form.     

On the fast reduction of a quasiseparable matrix to Hessenberg and tridiagonal forms

In this paper we design a fast new algorithm for reducing an N × N quasiseparable matrix to upper Hessenberg form via a sequence of N − 2 unitary transformations. The new reduction is especially useful when it is followed by the QR algorithm to obtain a complete set of eigenvalues of the original matrix. In particular, it is shown that in a number of cases some recently devised fast adaptations of the QR method for quasiseparable matrices can benefit from using the proposed reduction as a preprocessing step, yielding lower cost and a simplification of implementation.     

Krylov type subspace methods for matrix polynomials

We consider solving eigenvalue problems or model reduction problems for a quadratic matrix polynomial Iλ² − Aλ − B with large and sparse A and B. We propose new Arnoldi and Lanczos type processes which operate on the same space as A and B live and construct projections of A and B to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem. We shall apply the new processes to solve eigenvalue problems and model reductions of a second order linear input-output system and discuss convergence properties. Our new processes are also extendable to cover a general matrix polynomial of any degree.     

The spectra of some trees and bounds for the largest eigenvalue of any tree

Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n_k−j+1 and d_k−j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (n_k = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d₁, d₂, …, d_k−1, d_k ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.     

The LU-factorization of totally positive matrices

An n × n real matrix A is an STP (strictly totally positive) matrix if all its minors are strictly positive. An n × n real triangular matrix A is a ΔSTP matrix if all its nontrivial minors are strictly positive. It is proved that A is an STP matrix iff A = LU where L is a lower triangular matrix, U is an upper triangular matrix, and both L and U are ΔSTP matrices. Several related results are proved. These results lead to simple proofs of many of the determinantal properties of STP matrices.     

Eigenvalue inequalities for convex and log-convex functions

We give a matrix version of the scalar inequality f(a + b) ? f(a) + f(b) for positive concave functions f on [0, ∞). We show that Choi’s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences. Some inequalities for log-convex functions are presented and a new arithmetic-geometric mean inequality for positive matrices is given. We also point out a simple proof of the Bhatia-Kittaneh arithmetic-geometric mean inequality.     

Spectral criteria for complementary triangular forms

In this paper we consider the following problem: Given two matricesA,Z∈? ^n×n , does there exist an invertiblen×n-matrixS such thatS ^?1 AS is an upper triangular matrix andS ^?1 ZS is a lower triangular matrix, and if so, what can be said about the order in which the eigenvalues ofA andZ appear on the diagonals of these triangular matrices? For special choices ofA andZ a complete solution is possible, as has been shown by several authors. Here we follow a lead, provided by Shmuel Friedland, who discussed the case where bothA andZ have at leastn-1 linearly independent eigenvectors, and we descibe the problem in terms of Jordan chains and left-Jordan chains for the matricesA, Z. The results give some insight in the question why certain classes of matrices (like the nonderogatory and the rank 1 matrices) allow for a detailed solution of the problems described above; for some of these classes the result of this analysis is presented here for the first time.