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:
where denotes a matrix of constant coefficients, and is non-zero.
An example 3×3 polynomial matrix, degree 2:
We can express this by saying that for a
ringR, 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 determinant of a matrix polynomial with
Hermitianpositive-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