多项式矩阵的最小多项式的快速算法

——————————————————————————————————————————– Abstract In this paper, an efficient algorithm for computing the minimal polynomial of a polynomial matrix is presented. It determines the coefficient polynomials term by term from lower to higher degree. By using a random vector and randomly shifting, it requires no condition on the input matrix. Comparison with other algorithms in both theoretical complexity analysis and computational tests are given to show its effectiveness. Keywords: polynomial matrix; minimal polynomial; characteristic polynomial ———————————————————————————————————————————
1. Introduction
The minimal polynomial of a square matrix is a basic concept in linear algebra and its computation has important applications in automatic control and other fields. For example, to study the stability of the system with linear differential equation
−1 c0,0 I + c1,0 A0 + . . . + cn−1,0 An + An 0 = 0. 0
Multiplying both sides of the above equation with y from the right, we get
−1 that the matrix (y, A0 y, . . . , An y ) is nonsingular. 0
Let c(λ, x) = λn + cn−1 (x)λn−1 + . . . + c0 (x) be the characteristic polynomial of the matrix A(x), where ci (x) = ci,0 + ci,1 x + . . . + ci,di xdi . We define c ¯i,j (x) as c ¯i,j (x) = ci,0 + ci,1 x + . . . + ci,j xj , where ci,j = 0 (j > di ). Note that for any positive integer l, we have ci (x) ≡ c ¯i,l−1 (x) + ci,l xl (mod xl+1 ) ≡ c ¯i,l−1 (x) (mod xl ). By Caley-Hamilton Theorem, we have c(A(x), x) = 0. Hence, we have c0 (x)I + . . . + cn−1 (x)A(x)n−1 + A(x)n = 0. Looking at the constant part of the above matrix equation, we obtain
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An Efficient Algorithm for Computing Minimal Polynomials of Polynomial Matrices
Bo Yu and Yanyan Xu
Department of Applied Mathematics, Dalian University of Technology Dalian, Liaoning 116024, P.R.China yubo@dlut.edu.cn
then the characteristic polynomial of matrix A is given by c(λ) = λn + cn−1 λn−1 + . . . + cn . This is a fraction-free method and hence can also be applied to a polynomial matrix. It needs n3 (n − 1) polynomial multiplications when it is used to compute the characteristic polynomial of a polynomial matrix. The CHTB method ([6]): Let A(x) = A0 + A1 x + . . . + Ad xd be an n × n polynomial matrix. Compute first the eigenvalues λ1 , . . . , λn and eigenvectors v1 , . . . , vn of constant part A0 of matrix A(x) by some numerical method, and define the vector y = v1 + . . . + vn , then the author proved
= Ax + bu y = cx where A is an n × n matrix, x and b are column vectors, c is a row vector, u and y are scalars, one need to check if all eigenvalues of A have negative real parts. However, when A is of dimension 1
2
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Faddeev-Leverrier’s method ([3], [5]): Let A1 = A Ai ) (1 ≤ i ≤ n) c = − trace( i i Ai+1 = A(Ai + ci I ) (1 ≤ i ≤ n − 1)
dx dt
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bigger than 4, eigenvalues can not be computed exactly. One can compute the characteristic polynomial of A, and check if all roots of the characteristic polynomial have negative real parts by Routh Hurwitz stability criteria (see [2]). In fact, the roots of the minimal polynomial contains all of the roots of the characteristic polynomial, so it is enough to compute the minimal polynomial for recognizing if all eigenvalues of A have negative real parts. In some applications, eg., the stability problem above in case of A is a polynomial matrix in some unknowns, we need to compute the characteristic polynomial or the minimal polynomial of a polynomial matrix. For a constant matrix, there are several algorithms to compute its characteristic polynomial, and the major ones are Faddeev-Leverrier’s algorithm, Lagrange interpolation method, Hessenberg method, Danilevskii method and etc. (see [3], [1], [5], [8], [9], [10], [13]). As being a fraction-free method, Faddeev-Leverrier’s method can be applied directly to polynomial matrices. In [7], a fraction-free Danilevskii method was given. In [6], an efficient method based on Caley-Hamilton Theorem (CHTB method) for computing characteristic polynomial of a polynomial matrix was given. It needs the constant part of the input matrix to have n different eigenvalues. In [12], a Cayley-Hamilton based method with an artificial constant matrix (CHACM method) was given to release this restriction in CHTB method. In the following, we give a brief review of Faddeev-Leverrier’s method, CHTB method and CHACM method. Before that, we list some notations which we will use in the rest of this paper: (y1 , y2 , . . . , yn ): the matrix with column vectors y1 , y2 , . . . , yn . degx (g (x)): the total degree of polynomial g (x) about x. I : the identity matrix. < y >k : the vector whose entries are coefficients of (x − x0 )k in y . [M ]l : the lth column of matrix M . trace(M ): the trace of matrix M . rank(M ): the rank of matrix M . [G]i1 ,...,ir : the matrix whose 1st, . . . ,rth row are in order the i1 th, . . . , ir th row of matrix G. [b]i1 ,...,ir : the vector whose the 1st, . . . , rth entries are in order the i1 th, . . . , ir th entries of b.