Central polynomial explained

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 and y.

See also

References

. Edward W. Formanek . The polynomial identities and invariants of n×n matrices . 0714.16001 . Regional Conference Series in Mathematics . 78 . Providence, RI . . 1991 . 0-8218-0730-7.

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