Generalized even and odd totally positive matrices

A generalization of the definition of an oscillatory matrix based on the theory of cones is given in this paper. The positivity and simplicity of all the eigenvalues of a generalized oscillatory matrix are proved. Classes of generalized even and odd oscillatory matrices are introduced. Spectral properties of the obtained matrices are studied. Criteria of generalized even and odd oscillation are given. Examples of generalized even and odd oscillatory matrices are presented.     

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.     

Perturbing non-real eigenvalues of non-negative real matrices

Let σ=(ρ,b+ic,b-ic,λ₄,…,λ_n) be the spectrum of an entry non-negative matrix and t?0. Laffey [T. J. Laffey, Perturbing non-real eigenvalues of nonnegative real matrices, Electron. J. Linear Algebra 12 (2005) 73-76] has shown that σ=(ρ+2t,b-t+ic,b-t-ic,λ₄,…,λ_n) is also the spectrum of some nonnegative matrix. Laffey (2005) has used a rank one perturbation for small t and then used a compactness argument to extend the result to all nonnegative t. In this paper, a rank two perturbation is used to deduce an explicit and constructive proof for all t?0.     

Relative perturbation bounds for the unitary polar factor

LetB be anm×n (m≥n) complex (or real) matrix. It is known that there is a uniquepolar decomposition B=QH, whereQ*Q=I, then×n identity matrix, andH is positive definite, providedB has full column rank. Existing perturbation bounds suggest that in the worst case, for complex matrices the change inQ be proportional to the reciprocal ofB's least singular value, or the reciprocal of the sum ofB's least and second least singular values if matrices are real. However, there are situations where this unitary polar factor is much more accurately determined by the data than the existing perturbation bounds would indicate. In this paper the following question is addressed: how much mayQ change ifB is perturbed to $\tilde B = D_1^* BD_2 $ , whereD ₁ andD ₂ are nonsingular and close to the identity matrices of suitable dimensions? It is shown that for a such kind of perturbation, the change inQ is bounded only by the distances fromD ₁ andD ₂ to identity matrices and thus is independent ofB's singular values. Such perturbation is restrictive, but not unrealistic. We show how a frequently used scaling technique yields such a perturbation and thus scaling may result in better-conditioned polar decompositions.     

Positive semidefinite quadratic forms on unitary matrices

Let

     

On generators of bounded ratios of minors for totally positive matrices

We provide a method for factoring all bounded ratios of the form

     

The algebraic connectivity of graphs under perturbation

In this paper, we investigate how the algebraic connectivity of a connected graph behaves when the graph is perturbed by separating or grafting an edge.     

Fast accurate eigenvalue methods for graded positive definite matrices

Summary. Let where is a positive definite matrix and is diagonal and nonsingular. We show that if the condition number of is much less than that of then we can use algorithms based on the Cholesky factorization of to compute the eigenvalues of to high relative accuracy more efficiently than by Jacobi's method. The new methods are generally slower than tridiagonalization methods (which do not deliver the eigenvalues to maximal relative accuracy) but can be up to 4 times faster when the condition number of is very large. Received April 13, 1995     

Polynomial numerical hulls of some normal matrices

In this note, polynomial numerical hulls of matrices of the form _A1⊕i_A2$A_1\oplus i{A}_2$, where _A1$A_1$ and _A2$A_2$ are Hermitian, are characterized.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

On the numbers of positive entries of reducible nonnegative matrices

Let RM(n,d) be the class {A∣A is an n×n reducible nonnegative matrix and the greatest common divisor of the lengths of all cycles in D(A) is d}, where D(A) is the associated digraph of A. In this paper we determine the set of numbers of positive entries of A for A∈RM(n,d). We also characterize the reducible nonnegative matrices with the maximum and minimum numbers of positive entries.     

A note on upper bounds on the cp-rank

Hanna and Laffey gave an upper bound on the cp-rank of a completely positive matrix, in terms of its rank and the number of zeros in a full rank principal submatrix. This bound, for the case that the matrix is positive, was improved by Barioli and Berman. In this paper a new straightforward proof of both results is given, and the same approach is used to improve Hanna and Laffey’s bound in the case that the matrix has a zero entry.     

Bounds on eigenvalues of the Hadamard product and the Fan product of matrices

We prove an upper bound for the spectral radius of the Hadamard product of nonnegative matrices and a lower bound for the minimum eigenvalue of the Fan product of M-matrices. These improve two existing results.     

Some new bounds on the spectral radius of matrices

A new lower bound on the smallest eigenvalue τ(AB) for the Fan product of two nonsingular M-matrices A and B is given. Meanwhile, we also obtain a new upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B. These bounds improve some results of Huang (2008) [R. Huang, Some inequalities for the Hadamard product and the Fan product of matrices, Linear Algebra Appl. 428 (2008) 1551-1559].     

Extremal numbers of positive entries of imprimitive nonnegative matrices

We determine the maximum and minimum numbers of positive entries of imprimitive nonnegative matrices with a given imprimitivity index. One application of the results is to estimate the imprimitivity index by the number of positive entries. The proofs involve the study of a cyclic quadratic form. This completes a research initiated by Lewin in 1990.     

Notes on Hilbert and Cauchy matrices

Inspired by examples of small Hilbert matrices, the author proves a property of symmetric totally positive Cauchy matrices, called AT-property, and consequences for the Hilbert matrix.     

The geometric mean and B-loop quasigroups of the convex cone of positive definite matrices

In this paper we consider the convex cone of positive definite matrices as algebraic system equipped with geometric mean and B-loop from the standard matrix polar decomposition. Some algebraic structures of these quasigroups are investigated in the context of matrix theory. In particular, their autotopism groups are completely determined: they are isomorphic to the group of positive real numbers.Received: 28 April 2004     

The constrained inverse eigenvalue problem and its approximation for normal matrices

In this paper, the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered. The solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented, and the expression of the solution for the optimal approximation problem is obtained.     

Structure of Hiai-Petz parametrized geometry for positive definite matrices

Recently Hiai-Petz (2009) [10] discussed a parametrized geometry for positive definite matrices with a pull-back metric for a diffeomorphism to the Euclidean space. Though they also showed that the geodesic is a path of operator means, their interest lies mainly in metrics of the geometry. In this paper, we reconstruct their geometry without metrics and then we show their metric for each unitarily invariant norm defines a Finsler one. Also we discuss another type of geometry in Hiai and Petz (2009) [10] which is a generalization of Corach-Porta-Recht’s one [3].