In mathematics, a matrix polynomial is a polynomial with
square matrices as variables. Given an ordinary, scalar-valued polynomial
:
this polynomial evaluated at a matrix
is
:
where
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
has the same dimension as
.
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'').
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
. The
Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix
itself, the result is the zero matrix:
. An polynomial ''annihilates''
if
;
is also known as an ''
annihilating polynomial''. Thus, the characteristic polynomial is a polynomial which annihilates
.
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
; this polynomial is the
minimal polynomial. Any polynomial which annihilates
(such as the characteristic polynomial) is a multiple of the minimal polynomial.
It follows that given two polynomials
and
, we have
if and only if
:
where
denotes the
th derivative of
and
are the
eigenvalues of
with corresponding indices
(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 ...
,
:
:
:
:
If
is nonsingular one can evaluate the expression for the sum
.
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