Simultaneous diagonalization of three real symmetric matrices

We formulate and prove necessary and sufficient conditions of simultaneous diagonalization of three real symmetric matrices of regular pencil. The conditions are algebraic and consist, in particular, of two spectral requirements and one matrix equality. For the degenerate matrix pencil we suggest an approach that allows reducing of the analysis to a regular pencil. With the use of obtained theorems we investigate a decomposition of linear gyroscopic system into subsystems of an order not higher than two and the stability of trivial solution to a system.     

Simultaneous block triangularization and block diagonalization of sets of matrices

We consider the problem of simultaneously putting a set of square matrices into the same block upper triangular form with a similarity transformation, and obtain a result linking the size of the largest block to polynomial identities. This is used to yield a new proof of a theorem of Watters [20] which gives a necessary and sufficient condition for a set of matrices to be simultaneously, unitarily similar to block diagonal matrices with blocks of sizes one or two.     

Simultaneous diagonalization of rectangular matrices

A matrix D is said to be diagonal if its (i,j)th element is null whenever i and j are unequal. For a set {A_θ} of matrices A_θ of the same order, the paper gives necessary and sufficient conditions for nonsingular matrices S and T to exist, such that SA_θT = D_θ is diagonal for each matrix A_θ in the set.     

Fast block diagonalization of k-tridiagonal matrices

In the present paper, we give a fast algorithm for block diagonalization of k-tridiagonal matrices. The block diagonalization provides us with some useful results: e.g., another derivation of a very recent result on generalized k-Fibonacci numbers in [M.E.A. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461]; efficient (symbolic) algorithm for computing the matrix determinant.     

Extreme eigenvalues of real symmetric Toeplitz matrices

We exploit the even and odd spectrum of real symmetric Toeplitz matrices for the computation of their extreme eigenvalues, which are obtained as the solutions of spectral, or secular, equations. We also present a concise convergence analysis for a method to solve these spectral equations, along with an efficient stopping rule, an error analysis, and extensive numerical results.

     

A bandstructure-invariant deflation for real symmetric matrices

By means of an eigenvector and eigenvalue of a real symmetric matrix A, a unitary matrix U is constructed such that U¹AU deflates A and, moreover, the transformation preserves the bandstructure.     

Spectral functions for real symmetric Toeplitz matrices

We derive separate spectral functions for the even and odd spectra of a real symmetric Toeplitz matrix, which are given by the roots of those functions. These are rational functions, also commonly referred to as secular functions. Two applications are considered: spectral evolution as a function of one parameter and the computation of eigenvalues.     

On bounds for eigenvalues of real symmetric matrices

Let A=(a_ij) be a real symmetric matrix of order n. We characterize all nonnegative vectors x=(x₁,...,x_n) and y=(y₁,...,y_n) such that any real symmetric matrix B=(b_ij), with b_ij=a_ij, i≠jhas its eigenvalues in the union of the intervals [b_ij?y_i, b_ij+ x_i]. Moreover, given such a set of intervals, we derive better bounds for the eigenvalues of B using the 2n quantities {b_ii?y, b_ii+x_i}, i=1,..., n.     

Linear combinations of Hermitian and real symmetric matrices

This paper, by purely algebraic and elementary methods, studies useful criteria under which the quadratic forms x′Ax and x′Bx, where A,B are n × n symmetric real matrices and x′=(x₁,x₂, …,x_n)≠(0,0,0,0, …,0), can vanish simultaneously and some real linear combination of A,B can be positive definite. Analogous results for hermitian matrices have also been discussed. We have given sufficient conditions on m real symmetric matrices so that some real linear combination of them can be positive definite.     

On Lyapunov scaling factors of real symmetric matrices

A necessary and sufficient condition for the identity matrix to be the unique Lyapunov scaling factor of a given real symmetric matrix A is given. This uniqueness is shown to be equivalent to the uniqueness of the identity matrix as a scaling D for which the kernels of A and AD are identical.     

On real matrices with positive definite symmetric component

Inequalities concerning real square matrices A with positive definite symmetric component A+A^*are derived from certain inertia relations which hold for any complex (not necessarily real) square matrices A with positive definite

A+A^*     

On latently real matrices and block quaternions

Let a complex n × n matrix A be unitarily similar to its entrywise conjugate matrix [`(A)] \bar{A} . If in the relation [`(A)] = P^*AP \bar{A} = {P^*}AP the unitary matrix P can be chosen symmetric (skew-symmetric), then A is called a latently real matrix (respectively, a generalized block quaternion). The differences in the systems of elementary divisors of these two matrix classes are found that explain why latently real matrices can be made real via unitary similarities, whereas, in general, block quaternions cannot. Bibliography: 5 titles.     

Bounds for eigenvalues of symmetric block Jacobi scaled matrices

The paper presents upper bounds for the largest eigenvalue of a block Jacobi scaled symmetric positive-definite matrix which depend only on such parameters as the block semibandwidth of a matrix and its block size. From these bounds we also derive upper bounds for the smallest eigenvalue of a symmetric matrix with identity diagonal blocks. Bibliography: 4 titles. Translated by L. Yu. Kolotilina. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 18–25.     

On the simultaneous tridiagonalization of two symmetric matrices

We discuss congruence transformations aimed at simultaneously reducing a pair of symmetric matrices to tridiagonal–tridiagonal form under the very mild assumption that the matrix pencil is regular. We outline the general principles and propose a unified framework for the problem. This allows us to gain new insights, leading to an economical approach that only uses Gauss transformations and orthogonal Householder transformations. Numerical experiments show that the approach is numerically robust and competitive.     

一类特殊的对称的块循环矩阵的特征值

In this paper, the spectrum and characteristic polynomial for a special kind of symmetric block circulant matrices are given.