Spectral inequalities and equalities involving products of matrices (original) (raw)

Reducible spectral theory with applications to the robustness of matrices

Let a ⊕ b = max(a, b) and a ⊗ b = a + b for a, b ∈ R := R ∪ {−∞}. By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕, ⊗), extended to matrices and vectors. The symbol A k stands for the kth max-algebraic power of a square matrix A. Let us denote by ε the max-algebraic "zero" vector, all the components of which are −∞. The max-algebraic eigenvalue-eigenvector problem is the following: Given A ∈ R n×n , find all λ ∈ R and

Spectra of principal submatrices of nonnegative matrices

Linear Algebra and its Applications, 2008

Let A be an n × n nonnegative matrix with the spectrum (λ 1 , λ 2 ,. .. , λ n) and let A 1 be an m × m principal submatrix of A with the spectrum (μ 1 , μ 2 ,. .. , μ m). In this paper we present some cases where the realizability of (μ 1 , μ 2 ,. .. , μ m , ν 1 , ν 2 ,. .. , ν s) implies the realizability of (λ 1 , λ 2 ,. .. , λ n , ν 1 , ν 2 ,. .. , ν s) and consider the question of whether this holds in general. In particular, we show that the list

Linear Algebra and its Applications 363 (2003) 65--80

We present new comparison theorems for the spectral radii of matrices arising from splittings of different matrices under nonnegativity assumptions. Our focus is on establishing strict inequalities of the spectral radii without imposing strict inequalities of the matrices, but we also obtain new results for nonstrict inequalities of the spectral radii. We emphasize two different approaches, one combinatorial and the other analytic and discuss their merits in the light of the results obtained. We try to get fairly general results and indicate by counter-examples that some of our hypotheses cannot be relaxed in certain directions.

A criterion for reducibility of matrices

The problem of existence and characterization of non-trivial reducing subspaces for a given matrix is studied employing some basic tools of multilinear algebra. A criterion for reducibility of a single matrix is obtained which is also extended to the case of simultaneous reduction of two or more matrices.

On eigenvalues of rectangular matrices

2007

Given a (k+1)(k+1)(k+1)-tuple A,B1,...,BkA, B_1,...,B_kA,B1,...,Bk of (mtimesn)(m\times n)(mtimesn)-matrices with mlenm\le nmlen we call the set of all kkk-tuples of complex numbers la1,...,lak\{\la_1,...,\la_k\}la1,...,lak such that the linear combination A+la1B1+la2B2+...+lakBkA+\la_1B_1+\la_2B_2+...+\la_kB_kA+la_1B_1+la2B2+...+lakBk has rank smaller than mmm the {\it eigenvalue locus} of the latter pencil. Motivated primarily by applications to multi-parameter generalizations of the Heine-Stieltjes spectral problem, see \cite{He} and \cite{Vol}, we study a number of properties of the eigenvalue locus in the most important case k=n−m+1k=n-m+1k=n−m+1.

On the second real eigenvalue of nonnegative and Z-matrices

1999

We give bounds for the second real eigenvalue of nonnegative matrices and Z{matrices. Furthermore, we establish upper bounds for the maximal spectral radii of principal submatrices of nonnegative matrices. Using this bounds we prove that our inequality for the second real eigenvalue of the incidence matrix of a connected regular graph improves a well known bound for the second eigenvalue using Cheeger's inequality.