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:

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

where $A(i)$ denotes a matrix of constant coefficients, and $A(p)$ is non-zero.

An example 3×3 polynomial matrix, degree 2:

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

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.{{Cite journal|last1=Friedland|first1=S.|last2=Melman|first2=A.|date=2020|title=A note on Hermitian positive semidefinite matrix polynomials|journal=Linear Algebra and Its Applications|language=en|volume=598|pages=105–109|doi=10.1016/j.laa.2020.03.038}}

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

{{cite book |first=E.V. |last=Krishnamurthy |title=Error-free Polynomial Matrix computations |publisher=Springer |date=1985 |doi=10.1007/978-1-4612-5118-7 |isbn=978-1-4612-9572-3 |oclc=858879932}}

Category:Matrices (mathematics)

Category:Polynomials