Polynomial Matrix

In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices.

A univariate polynomial matrix ''P'' of degree ''p'' is defined as:

:$P = \sum_^p A(n)x^n = A(0)+A(1)x+A(2)x^2+ \cdots +A(p)x^p$

where $A(i)$ denotes a matrix of constant coefficients, and $A(p)$ is non-zero.
An example 3×3 polynomial matrix, degree 2:

:$P=\begin 1 & x^2 & x \\ 0 & 2x & 2 \\ 3x+2 & x^2-1 & 0 \end =\begin 1 & 0 & 0 \\ 0 & 0 & 2 \\ 2 & -1 & 0 \end +\begin 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3 & 0 & 0 \endx+\begin 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \endx^2.$

We can express this by saying that for a ring ''R'', the rings M_n(R and
(M_n(R)) /math> are isomorphic.

Properties

*A polynomial matrix over a field with determinant equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function.
*The roots of a polynomial matrix over the complex numbers are the points in the complex plane where the matrix loses rank.
*The determinant of a matrix polynomial with Hermitian positive-definite (semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients.

Note that polynomial matrices are ''not'' to be confused with monomial matrices, which are simply matrices with exactly one non-zero entry in each row and column.

If by λ we denote any element of the field over which we constructed the matrix, by ''I'' the identity matrix, and we let ''A'' be a polynomial matrix, then the matrix λ''I'' − ''A'' is the characteristic matrix of the matrix ''A''. Its determinant, , λ''I'' − ''A'', is the characteristic polynomial of the matrix ''A''.

References

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Matrices (mathematics)
Polynomials

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