Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series

Abstract

Given a square matrix A, Brauer's theorem [Duke Math. J. 19 (1952), [75][76][77][78][79][80][81][82][83][84][85][86][87][88][89][90][91] shows how to modify one single eigenvalue of A via a rank-one perturbation, without changing any of the remaining eigenvalues. We reformulate Brauer's theorem in functional form and provide extensions to matrix polynomials and to matrix Laurent series A(z) together with generalizations to shifting a set of eigenvalues. We provide conditions under which the modified function A(z) has a canonical factorization A(z) = U (z) L(z −1 ) and we provide explicit expressions of the factors U (z) and L(z). Similar conditions and expressions are given for the factorization of A(z −1 ). Some applications are discussed. * Work supported by Gruppo Nazionale di Calcolo Scientifico (GNCS) of INdAM, and by the PRA project "Mathematical models and computational methods for complex networks" funded by the University of Pisa. inverse eigenvalue problem [10], [21], to accelerating the convergence of iterative methods for solving matrix equations in particular quadratic equations [20], equations related to QBD Markov chains [17], [14], to M/G/1-type Markov chains [5], [3], or for algebraic Riccati equations in [15], [16], [19], and [2].In this paper we revisit Brauer's theorem in functional form and generalize it to a non-degenerate n × n matrix function A(z) analytic in the open annulusHere, non-degenerate means that det A(z) is not identically zero. Throughout the paper we assume that all the matrix functions are nondegenerate.Let λ ∈ A r1,r2 be an eigenvalue of A(z), i.e., such that det A(λ) = 0 and u ∈ C n , u = 0, be a corresponding eigenvector, i.e., such that A(λ)u = 0. We prove that for any v ∈ C n such that v * u = 1 and for any µ ∈ C, the functionwhere v * denotes the transpose conjugate of v. Moreover, A(z) has all the eigenvalues of A(z), except for λ which, if µ ∈ A r1,r2 , is replaced by µ, otherwise is removed. We provide explicit expressions which relate the coefficients A i of A(z) = i∈Z z i A i to those of A(z).This transformation, called right shift, is obtained by using a right eigenvector u of A(z). A similar version, called left shift, is given by using a left eigenvector v of A(z). Similarly, a combination of the left and the right shift, leads to a transformation called double shift, which enables one to shift a pair of eigenvalues by operating to the right and to the left of A(z).This result, restricted to a matrix Laurent polynomial A(z) = k −h z i A i , provides a shifted function which is still a matrix Laurent polynomial A(z) = k i=−h z i A i . In the particular case where A(z) = zI − A, we find that A(z) = zI − A is such that A is the matrix given in the classical theorem of Brauer [9]. Therefore, these results provide an extension of Brauer's Theorem 1.1 to matrix Laurent series and to matrix polynomials.A further generalization is obtained by performing the right, left, or double shift simultaneously to a packet of eigenvalues once we are given an invariant (right or le...

Efficient Ehrlich–Aberth iteration for finding intersections of interpolating polynomials and rational functions

“…The proof is just more technical but uses the same ideas, so we omit it. The same result can be obtained by relying on the theorem in [5] twice, first moving the Jordan chain at infinity to some finite point and then moving it to zero.…”

“…Notice, however, the explicit computation of the matrix M is not needed and one can perform the iteration by solving a certain number of linear systems. In our case, however, B is singular 3 , so we make use of Brauer's theorem, which is a simple yet powerful tool that allows one to move a specified eigenvalue of a matrix [8] and, more generally, of matrix functions expressed as Laurent series [5]. In our case we are interested in shifting an entire Jordan chain from the infinite eigenvalue to a zero one, such that it will not interfere with the power iteration and estimation of the dominant finite eigenvalue.…”

“…This is a generalization of the original one in [8], and a particular case of [5]. Our formulation allows to transparently deal with the shift of infinity eigenvalues to 0, which is not achievable directly with the formulations in [5,8]. To achieve this, we identify the eigenvalues of the pencil with the projective points in P 1 (C).…”

“…The integrand of Theorem 3.4 is also called the logarithmic derivative of f (λ). We notice that it is nothing else than the inverse of Newton's correction f (λ)/f (λ) evaluated at the point λ, according to (5). In the following we will show how to evaluate this function in O(n) flops.…”

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“…This can be readily proved by considering the determinants det P (z) = det P (z) det S(z) and their degrees. This formulation of shifting as multiplication by a suitable matrix polynomial has been suggested recently in [BM15].…”