{{inline |date=May 2024}} In algebra, a '''central polynomial''' for ''n''-by-''n'' matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever it is evaluated at ''n''-by-''n'' matrices. That such polynomials exist for any square matrices was discovered in 1970 independently by Formanek and Razmyslov. The term "central" is because the evaluation of a central polynomial has the image lying in the center of the matrix ring over any commutative ring. The notion has an application to the theory of polynomial identity rings.
Example: <math>(xy - yx)^2</math> is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that <math>(xy - yx)^2 = -\det(xy - yx)I</math> for any 2-by-2-matrices ''x'' and ''y''.
== See also == *Generic matrix ring
== References == *{{cite book | last=Formanek | first=Edward |authorlink= Edward W. Formanek | title=The polynomial identities and invariants of ''n''×''n'' matrices | zbl=0714.16001 | series=Regional Conference Series in Mathematics | volume=78 | location=Providence, RI | publisher=American Mathematical Society | year=1991 | isbn=0-8218-0730-7}} *{{cite web|last=Artin|first=Michael|title=Noncommutative Rings|url=http://math.mit.edu/~etingof/artinnotes.pdf|year=1999|location=V. 4.}}
Category:Ring theory
{{polynomial-stub}}