Editing Polynomial (section)

=== Matrix polynomials ===
{{Main|Matrix polynomial}}
A [[matrix polynomial]] is a polynomial with [[square matrix|square matrices]] as variables.<ref>{{cite book |title=Matrix Polynomials |volume=58 |series=Classics in Applied Mathematics |first1=Israel |last1=Gohberg |first2=Peter |last2=Lancaster |first3=Leiba |last3=Rodman |publisher=[[Society for Industrial and Applied Mathematics]] |location=Lancaster, PA |year=2009 |orig-year=1982 |isbn=978-0-89871-681-8 |zbl=1170.15300}}</ref> Given an ordinary, scalar-valued polynomial
<math display="block">P(x) = \sum_{i=0}^n{ a_i x^i} =a_0 + a_1 x+ a_2 x^2 + \cdots + a_n x^n, </math>
this polynomial evaluated at a matrix ''A'' is
<math display="block">P(A) = \sum_{i=0}^n{ a_i A^i} =a_0 I + a_1 A + a_2 A^2 + \cdots + a_n A^n,</math>
where ''I'' is the [[identity matrix]].{{sfn|Horn|Johnson|1990|p=36}}

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]] ''M<sub>n</sub>''(''R'').