Polynomial matrix

Not to be confused with matrix polynomial.

In mathematics, a polynomial matrix or sometimes matrix polynomial is a matrix whose elements are univariate or multivariate polynomials. A λ-matrix is a matrix whose elements are polynomials in λ.

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. Thus a polynomial matrix is the matrix-equivalent of a polynomial, with each element of the matrix satisfying the definition of a polynomial of degree p.

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

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