A polynomial matrices or matrix polynomial is a matrix whose elements are univariate or multivariate polynomials.A univariate polynomial matrix P of degree p is define like:sum_(n=0)^pA(n)x^(n)=A(0)+A(1)x+A(2)x^(2)+....+A(p)x^(p) where A(p) is non-zero and A(i) indicate a matrix of constant coefficients. Hence a polynomial matrix is the matrix-equivalent of a polynomial, by means of every one element of the matrix satisfying the classification of a polynomial of degree p.
Properties of polynomial matrices
A polynomial matrix in excess of a field with determinant equivalent to a non-zero constant is called unimodular, and have an inverse, which is also a polynomial matrix.
Note, that the simply scalar unimodular polynomials are polynomials of degree 0 - nonzero constants, for the reason that an inverse of an arbitrary polynomial of high degree is a rational function.
The roots of a polynomial matrix in excess of the complex numbers are the points in the complex plane wherever the matrix loses rank.
Characteristic polynomial of a product of two matrices
If A and B are two square n×n matrices then,attribute polynomials of AB and BA match:
PAB(t)=PBA(t).
If A is m×n-matrix and B is n×m matrices such that m
PAB(t)=tn-mPAB(t)
Polynomial in t and in the entry of A and B is a general polynomial identity. It consequently suffice to verify it on an open set ofparameter value in the complex numbers.
The tuples (A,B,t) wherever A is an invertible complex n by n matrix,
B is any complex n by n matrix,
and t is any complex number since an open set in complex space of dimension 2n2 + 1. When A is non-singular our result follow from the fact that AB and BA are similar:
BA=A-1(AB)A.
Example 1:
An example the 3x3 polynomial matrices
P=[[1,x^(2),x],[0,2x,2],[8x+2,x^2-1,0]]
=[[1,0,0],[0,0,2],[2,-1,0]]+[[0,0,1],[0,2,0],[3,0,0]]x+[[0,1,0],[0,0,0],[0,1,0]]xx^(2)
Example 2:
Find the eign value of given polynomial matrices
P=[[3,3],[0,6]]
The polynomial has the characteristic equation
0=det(P-λI)
=det[[3-lambda,3],[0,6-lambda]]
18-6lambda -3lambda + λ2
18-9lambda +18
λ2-3λ-6λ+18
λ(λ-3)-6(λ-3)
(λ-3)(λ-6)
λ=3,andλ=6
The eigenvalues of these matrices are 3,6
Example 3: Find the product of the given matrices M1=[[1,2],[3,4]] and M2=[[8,3],[2,7]]
The given polynomial is
M1=[[1,2],[3,4]]
M2=[[8,3],[2,7]]
The product of the given matrices M1 and M2 =M1xM2
M1xM2 =[[1,2],[3,4]]xx [[8,3],[2,7]]
The product of the given matrices is=[[12,17],[32,37]]
