Matrix polynomial

In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial
: $P(x) = \sum_^n =a_0 + a_1 x+ a_2 x^2 + \cdots + a_n x^n,$
this polynomial evaluated at a matrix ''A'' is
:$P(A) = \sum_^n =a_0 I + a_1 A + a_2 A^2 + \cdots + a_n A^n,$
where ''I'' is the identity matrix.

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 ''M_n''(''R'').

Characteristic and minimal polynomial

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.

It follows that given two polynomials ''P'' and ''Q'', we have $P(A) = Q(A)$ if and only if
:$P^(\lambda_i) = Q^(\lambda_i) \qquad \text j = 0,\ldots,n_i-1 \text i = 1,\ldots,s,$
where $P^$ denotes the ''j''th 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).

Matrix geometrical series

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

:$S=I+A+A^2+\cdots +A^n$
:$AS=A+A^2+A^3+\cdots +A^$
:$(I-A)S=S-AS=I-A^$
:$S=(I-A)^(I-A^)$

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

See also

* Latimer–MacDuffee theorem
* Matrix exponential
* Matrix function

Notes

References

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{{DEFAULTSORT:Matrix Polynomial
Matrix theory
Polynomials