Central polynomial

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: (xy - yx)^2 is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that (xy - yx)^2 = -\det(xy - yx)I for any 2-by-2-matrices x, y.

See also

References

This article is issued from Wikipedia - version of the Tuesday, February 03, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.