In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical method calculates successive square roots. In this article, a new algorithm is presented that relies on the computation of successive matrix exponentials. Conver...

Using the block vec matrix, I give a necessary and sufficient condition for factorization of a matrix into the Kronecker product of two other matrices. As a consequence, I obtain an elementary algorithmic procedure to decide whether a matrix has a square root for the Kronecker product. Introduction My statistician colleague, J.E. Chacón, asked me how to decide if a real given matrix A has a squ...

We consider two relations between Fisher's information matrix of a stationary ARMA (autoregressive moving average) process and Sylvester's resultant matrix. One is based on the Wald test statistic for testing common roots of the AR and MA polynomials of an ARMA process, and the other one is established by using the structure of Fisher's information matrix. It turns out that the latter is also a...

We estabish an analog of the Poincare-Cauchy separation theorem for normal matrices in terms of majorization. Moreover, we present a solution to the inverse spectral problem (Borgtype result) for a normal matrix. Using this result we essentially generalize and complement the known Gauss–Lucas theorem on the geometry of the roots of a complex polynomial and of its derivative. In turn the last re...

We present methods to compute verified square roots of a square matrix A. Given an approximation X to the square root, obtained by a classical floating point algorithm, we use interval arithmetic to find an interval matrix which is guaranteed to contain the error of X. Our approach is based on the Krawczyk method which we modify in two different ways in such a manner that the computational comp...

Abstract. If an n× n complex matrix A is nonsingular, then for every integer m > 1, A has an mth root B, i.e., Bm = A. In this paper, we present a new simple proof for the Jordan canonical form of the root B. Moreover, a necessary and sufficient condition for the existence of mth roots of a singular complex matrix A is obtained. This condition is in terms of the dimensions of the null spaces of...

Example 1. The matrix ( 0 −1 1 0 ) in M2(R) is not diagonalizable, but it becomes diagonalizable in M2(C) since its characteristic polynomial splits with distinct roots in C[T ]. Example 2. The matrix ( 1 1 0 1 ) is not diagonalizable over any field. Indeed, its only eigenvalue is 1 and its only eigenvectors are scalar multiples of ( 1 0 ) , so there is never a basis of eigenvectors for this ma...

We study the circulant complex Hadamard matrices of order n whose entries are l-th roots of unity. For n = l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n = p+ q, l = pq with p, q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n, l.

A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis’s implicitly shifted QR algorithm. A companion matrix is an upper Hessenberg matrix that is unitary-plus-rankone, that is, it is the sum of a unitary matrix and a rank-one matrix. These properties are preserved by iterations of Franc...