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.[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

Friedland, S.; Melman, A. (2020). "A note on Hermitian positive semidefinite matrix polynomials". Linear Algebra and Its Applications. 598: 105–109. doi:10.1016/j.laa.2020.03.038.

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

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[1] Friedland, S.; Melman, A. (2020). "A note on Hermitian positive semidefinite matrix polynomials". Linear Algebra and Its Applications. 598: 105–109. doi:10.1016/j.laa.2020.03.038.