Matrix polynomial

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In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial

this polynomial evaluated at a matrix A is

where I is the identity matrix.[1]

A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring Mn(R).

Characteristic and minimal polynomial[edit]

The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by . The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix A itself, the result is the zero matrix: . The characteristic polynomial is thus a polynomial which annihilates A.

There is a unique monic polynomial of minimal degree which annihilates A; this polynomial is the minimal polynomial. Any polynomial which annihilates A (such as the characteristic polynomial) is a multiple of the minimal polynomial.[2]

It follows that given two polynomials P and Q, we have if and only if

where denotes the jth derivative of P and are the eigenvalues of A with corresponding indices (the index of an eigenvalue is the size of its largest Jordan block).[3]

Matrix geometrical series[edit]

Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary geometric series,

If I − A is nonsingular one can evaluate the expression for the sum S.

See also[edit]

Notes[edit]

  1. ^ Horn & Johnson 1990, p. 36.
  2. ^ Horn & Johnson 1990, Thm 3.3.1.
  3. ^ Higham 2000, Thm 1.3.

References[edit]

  • Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) [1982]. Matrix Polynomials. Classics in Applied Mathematics. 58. Lancaster, PA: Society for Industrial and Applied Mathematics. ISBN 0-898716-81-0. Zbl 1170.15300.
  • Higham, Nicholas J. (2000). Functions of Matrices: Theory and Computation. SIAM. ISBN 089-871-777-9..
  • Horn, Roger A.; Johnson, Charles R. (1990). Matrix Analysis. Cambridge University Press. ISBN 978-0-521-38632-6..