For a nonsingular integer matrix A, we study the growth of the order of A modulo N . We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values o...

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, th...

In response to a question posed by João P. Silva, I demonstrate that an arbitrary 2n×2n real orthogonal antisymmetric matrix can be parameterized by n(n−1) continuous angular parameters. An algorithm is provided for constructing the general form for such a matrix. This algorithm is based on observation that Sp(n,R) ∩ O(2n) ∼= U(n) and employs the parameterization of the coset space SO(2n)/U(n)....

In recent years, an algebraic framework was introduced for the analysis of convergence of Schwarz methods for the solution of linear systems of the form Ax = b. Within this framework, additive and multiplicative Schwarz were shown to converge when the coefficient matrix A is a nonsingular M -matrix, or a symmetric positive definite matrix. In this paper, these results are extended to the case o...

A(t) is an hX« matrix with complex-valued elements which are measurable and bounded for ¿^0, and p is an «-vector with measurable, complex-valued elements. The norm of a vector (matrix) will be denoted by || -|| and is defined as the sum of the magnitudes of the elements. A vector (matrix) will be called bounded if its norm is bounded on i^O and convergent if its elements tend to finite limits ...

Let Ak, k ∈ N be a sequence of n×n complex valued matrices which converge to a matrix A. If A and each Ak is positive then the product AkAk−1...A2A1 ||AkAk−1...A2A1|| converges to a rank one matrix positive matrix uw, where u is a positive column eigenvector of A. If each Ak is nonsingular and A has exactly one simple eigenvalue λ of the maximal modulus with the corresponding eigenvector u, the...

Abstract This chapter considers the LU factorization of a general nonsymmetric nonsingular sparse matrix A . In practice, numerical pivoting for stability and/or ordering to limit fill-in in factors is often needed and computed then permuted PAQ Pivoting discussed Chapter 7 algorithms 8

A basic property of both polynomial and more general Tchebycheffian splines is that the associated B-spline collocation matrix has certain total positivity properties, and it is nonsingular if and only if well-known interlacing conditions hold. Using a technique from a recent paper of Mørken, we extend these results to L-splines. §

Linear system with M-matrices often appear in a wide variety of areas. In this work, we propose a new preconditioner for solving the system with nonsingular M-matrix. We show that our preconditioner increases the convergence rate of basic iterative methods. We also give a comparison between preconditioners with different parameters. Numerical results are also given.

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