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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 In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
. Note that P(A) has the same dimension as A. 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''). Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of linear transformations represented as matrices, most notably the Cayley–Hamilton theorem.


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. An polynomial ''annihilates'' A if p(A) = 0; p is also known as an '' annihilating polynomial''. Thus, the characteristic polynomial is a polynomial which annihilates A. There is a unique
monic polynomial In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one ...
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 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).


Matrix geometrical series

Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary
geometric series In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
, :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 In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exp ...
* Matrix function


Notes


References

* * . * . {{DEFAULTSORT:Matrix Polynomial Matrix theory Polynomials