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:

[math]\displaystyle{ P = \sum_{n=0}^p A(n)x^n = A(0)+A(1)x+A(2)x^2+ \cdots +A(p)x^p }[/math]

where [math]\displaystyle{ A(i) }[/math] denotes a matrix of constant coefficients, and [math]\displaystyle{ A(p) }[/math] is non-zero. An example 3×3 polynomial matrix, degree 2:

[math]\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. }[/math]

We can express this by saying that for a ring R, the rings [math]\displaystyle{ M_n(R[X]) }[/math] and [math]\displaystyle{ (M_n(R))[X] }[/math] 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.

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

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