Matrix polynomial

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Not to be confused with Polynomial matrix.

In mathematics, a matrix polynomial is a polynomial with matrices as variables. Given an ordinary, scalar-valued polynomial

P(x) = \sum_{i=0}^n{ a_i x^i} =a_0  + a_1 x+ a_2 x^2 + \cdots + a_n x^n,

this polynomial evaluated at a matrix A is

P(A) = \sum_{i=0}^n{ a_i A^i} =a_0 I + a_1 A + a_2 A^2 + \cdots + a_n A^n,

where I is the identity matrix.[1]

A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring Mn(R).

Characteristic and minimal polynomial[edit]

The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by p_A(t) = \det \left(tI - A\right). The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix A itself, the result is the zero matrix: p_A(A) = 0. The characteristic polynomial is thus a polynomial which annihilates A.

There is a unique monic polynomial of minimal degree which annihilates A; this polynomial is the minimal polynomial. Any polynomial which annihilates A (such as the characteristic polynomial) is a multiple of the minimal polynomial.[2]

It follows that that given two polynomials P and Q, we have  P(A) = Q(A) if and only if

 P^{(j)}(\lambda_i) = Q^{(j)}(\lambda_i) \qquad \text{for } j = 0,\ldots,n_i \text{ and } i = 1,\ldots,s,

where  P^{(j)} denotes the jth derivative of P and  \lambda_1, \dots, \lambda_s are the eigenvalues of A with corresponding indices  n_1, \dots, n_s (the index of an eigenvalue is the size of its largest Jordan block).[3]

Matrix geometrical series[edit]

Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary geometric series,

S=I+A+A^2+\cdots +A^n
AS=A+A^2+A^3+\cdots +A^{n+1}
(I-A)S=S-AS=I-A^{n+1}
S=(I-A)^{-1}(I-A^{n+1})

If I − A is nonsingular one can evaluate the expression for the sum S.

See also[edit]

Notes[edit]

  1. ^ Horn & Johnson 1990, p. 36.
  2. ^ Horn & Johnson 1990, Thm 3.3.1.
  3. ^ Higham 2000, Thm 1.3.

References[edit]