Polynomial matrix

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In mathematics, a polynomial matrix is a matrix whose elements are polynomials of a single variable.

A polynomial matrix A of degree p is defined as:

A = \sum_{n=0}^p A(n)x^n = A(0)+A(1)x+A(2)x^2+...+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


[edit] Properties

  • Any polynomial matrix with determinant ±1 is unimodular, and has an inverse which is also a polynomial matrix.
  • The roots of a polynomial matrix are the points in the complex plane where the matrix loses rank.


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.


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