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: [math]\displaystyle{ (xy - yx)^2 }[/math] is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that [math]\displaystyle{ (xy - yx)^2 = -\det(xy - yx)I }[/math] for any 2-by-2-matrices x and y.
See also
References
- Formanek, Edward (1991). The polynomial identities and invariants of n×n matrices. Regional Conference Series in Mathematics. 78. Providence, RI: American Mathematical Society. ISBN 0-8218-0730-7.
- Artin, Michael (1999). "Noncommutative Rings". V. 4.. http://math.mit.edu/~etingof/artinnotes.pdf.
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