Polynomial matrix explained

Polynomial matrix should not be confused with matrix polynomial.

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} _{n =0} A(n)x ^{n} = A( 0 ) + A( 1 )x + A( 2 )x ^{2+} … + 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{pmatrix} 1 & x ^{2} & x \\ 0 & 2 x & 2 \\ 3 x +2 & x ^{2-1} & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 2 \\ 2 & -1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3 & 0 & 0 \end{pmatrix}x + \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}x ^{2.}

We can express this by saying that for a ring R, the rings

M _{n(R[X])} and(M _{n(R))[X]} 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.^[1]

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 λI - A is the characteristic matrix of the matrix A. Its determinant, |λI - A| is the characteristic polynomial of the matrix A.

References

• E.V.Krishnamurthy, Error-free Polynomial Matrix computations, Springer Verlag, New York, 1985

Notes and References

1. Friedland. S.. Melman. A.. 2020. A note on Hermitian positive semidefinite matrix polynomials. Linear Algebra and Its Applications. en. 598. 105–109. 10.1016/j.laa.2020.03.038.