Polynomial matrix (original) (raw)

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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 A {\displaystyle A} $A$ of degree n {\displaystyle n} $n$ is defined as:[1] A ( x ) = ∑ i = 0 n A i x i = A 0 + A 1 x + A 2 x 2 + ⋯ + A n x n , {\displaystyle A(x)=\sum _{i=0}^{n}A_{i}x^{i}=A_{0}+A_{1}x+A_{2}x^{2}+\cdots +A_{n}x^{n},} $A(x)=\sum _{i=0}^{n}A_{i}x^{i}=A_{0}+A_{1}x+A_{2}x^{2}+\cdots +A_{n}x^{n},$ where A i {\displaystyle A_{i}} $A_{i}$ denotes a matrix of constant coefficients, and A n {\displaystyle A_{n}} $A_{n}$ is non-zero. An example 3×3 polynomial matrix, degree 2: A ( x ) = ( 1 x 2 x 0 2 x 2 3 x + 2 x 2 − 1 0 ) = ( 1 0 0 0 0 2 2 − 1 0 ) + ( 0 0 1 0 2 0 3 0 0 ) x + ( 0 1 0 0 0 0 0 1 0 ) x 2 . {\displaystyle A(x)={\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}.} $A(x)={\begin{pmatrix}1&x^{2}&x\\0&2x&2\\3x+2&x^{2}-1&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&0&2\\2&-1&0\end{pmatrix}}+{\begin{pmatrix}0&0&1\\0&2&0\\3&0&0\end{pmatrix}}x+{\begin{pmatrix}0&1&0\\0&0&0\\0&1&0\end{pmatrix}}x^{2}.$ We can express this by saying that for a ring R, the rings M n ( R [ X ] ) {\displaystyle M_{n}(R[X])} $M_{n}(R[X])$ and ( M n ( R ) ) [ X ] {\displaystyle (M_{n}(R))[X]} $(M_{n}(R))[X]$ are isomorphic.

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.

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 λ_I_ − A is the characteristic matrix of the matrix A. Its determinant, |λ_I_ − A| is the characteristic polynomial of the matrix A.

^ Gantmakher 1959, p. 130.

Gantmakher, Feliks Ruvimovich (1959). The Theory of Matrices - Volume 1. Providence, RI: Chelsea Publishing Company, Incorporated. ISBN 978-0-8218-1393-5.
Krishnamurthy, E.V. (1985). Error-free Polynomial Matrix computations. Springer. doi:10.1007/978-1-4612-5118-7. ISBN 978-1-4612-9572-3. OCLC 858879932.