最小多项式的性质及其应用

Qualities and Application of the Least Polynomial

给出矩阵A的最小多项式m(λ)的两个性质:(1)n阶矩阵A的全体实系数多项式所成的线性空间W的维数等于A的最 小多项式m(λ)的次数k;(2)对于次数大于零的任意多项式f(λ),f(A)为非退化的充分必要条件是f(λ)与m(λ)互素。并举例说 明了矩阵最小多项式在解决某些问题时的有效性。

In this paper, the two qualities of the least polynomial m (A) on matrix A are proposed. (1) If the total real coefficient polynomials on matrix A is the linear space W, then dimension of W equals number of times m (λ) .(2)If/(λ) is a arbitrary number of times polynomial, and (?)(f(λ) ) > 0,then f(A) is degenerative if and only if f(λ) and m(λ) is no common factor. Special examples are given to demonstrate the validity of using this two qualities for solving the problems.