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 ring R, the rings and are isomorphic.
Properties
[edit]- 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
[edit]- ^ 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.
- Krishnamurthy, E.V. (1985). Error-free Polynomial Matrix computations. Springer. doi:10.1007/978-1-4612-5118-7. OCLC 858879932.