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In mathematics, a matrix polynomial is a polynomial with
square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
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. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
. 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 In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...
''Mn''(''R'').


Characteristic and minimal polynomial

The
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
of a matrix ''A'' is a scalar-valued polynomial, defined by p_A(t) = \det \left(tI - A\right). The
Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies ...
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 In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\ ...
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
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s of ''A'' with corresponding indices n_1, \dots, n_s (the index of an eigenvalue is the size of its largest
Jordan block In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has t ...
).


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 the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each su ...
, :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

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Latimer–MacDuffee theorem The Latimer–MacDuffee theorem is a theorem in abstract algebra, a branch of mathematics. It is named after Claiborne Latimer and Cyrus Colton MacDuffee, who published it in 1933. Significant contributions to its theory were made later by Olga ...
*
Matrix exponential In mathematics, the matrix exponential is a matrix function on 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 exponential giv ...
*
Matrix function In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix, which is involved in th ...


Notes


References

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