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 is

where is the identity matrix.[1]

Note that has the same dimension as .

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).

Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of linear transformations represented as matrices, most notably the Cayley–Hamilton theorem.

Characteristic and minimal polynomial

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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   itself, the result is the zero matrix:  . An polynomial annihilates   if  ;   is also known as an annihilating polynomial. Thus, the characteristic polynomial is a polynomial which annihilates  .

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

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

 

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

Matrix geometrical series

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

 
 
 
 

If   is nonsingular one can evaluate the expression for the sum  .

See also

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Notes

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

References

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  • Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) [1982]. Matrix Polynomials. Classics in Applied Mathematics. Vol. 58. Lancaster, PA: Society for Industrial and Applied Mathematics. ISBN 978-0-898716-81-8. 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..