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Matrix polynomial

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

Contents

P ( x ) = ∑ i = 0 n a i x i = a 0 + a 1 x + a 2 x 2 + ⋯ + a n x n ,

this polynomial evaluated at a matrix A is

P ( A ) = ∑ i = 0 n a i A i = a 0 I + a 1 A + a 2 A 2 + ⋯ + 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 Mn(R).

Characteristic and minimal polynomial

The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by p A ( t ) = det ( t I − A ) . 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 ( j ) ( λ i ) = Q ( j ) ( λ i ) for  j = 0 , … , n i  and  i = 1 , … , s ,

where P ( j ) denotes the jth derivative of P and λ 1 , … , λ s are the eigenvalues of A with corresponding indices n 1 , … , 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 + ⋯ + A n A S = A + A 2 + A 3 + ⋯ + A n + 1 ( I − A ) S = S − A S = 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.

References

Matrix polynomial Wikipedia
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