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:
![{\displaystyle P=\sum _{n=0}^{p}A(n)x^{n}=A(0)+A(1)x+A(2)x^{2}+\cdots +A(p)x^{p))](https://wikimedia.org/api/rest_v1/media/math/render/svg/15227a59a3a39ce9df6eb49fd01627d8d9191e2d)
where
denotes a matrix of constant coefficients, and
is non-zero.
An example 3×3 polynomial matrix, degree 2:
![{\displaystyle 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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44c6fb61105c102dd2c7f7aac16cddd2ab7bb8d2)
We can express this by saying that for a ring R, the rings
and
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.