Central polynomial

In <a href="/facts/Abstract_algebra/YNNE1CNz">algebra</a>, a central polynomial for n-by-n <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> is a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> 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 <a href="/facts/Square_matrix/5TP9mNNn">square matrices</a> was discovered in 1970 independently by <a href="/facts/Edward_W._Formanek/JKjfD760">Formanek</a> and Razmyslov. The term "central" is because the evaluation of a central polynomial has the image lying in the <a href="/facts/Center_(ring_theory)/kJcejgkj">center</a> of the <a href="/facts/Matrix_ring/oj5HBr8B">matrix ring</a> over any <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>. The notion has an application to the theory of <a href="/facts/Polynomial_identity_ring/U0EPHo4I">polynomial identity rings</a>.
Example: 
 
 
 
 (
 x
 y
 −
 y
 x
 
 )
 
 2
 
 
 
 
 {\displaystyle (xy-yx)^{2}}
 
 is a central polynomial for 2-by-2-matrices. Indeed, by the <a href="/facts/Cayley%E2%80%93Hamilton_theorem/pXdJXWkR">Cayley–Hamilton theorem</a>, one has that 
 
 
 
 (
 x
 y
 −
 y
 x
 
 )
 
 2
 
 
 =
 −
 det
 (
 x
 y
 −
 y
 x
 )
 I
 
 
 {\displaystyle (xy-yx)^{2}=-\det(xy-yx)I}
 
 for any 2-by-2-matrices x and y.

Central polynomial open-in-new

Central polynomial