Trace identity explained

In mathematics, a trace identity is any equation involving the trace of a matrix.

Properties

Trace identities are invariant under simultaneous conjugation.

Uses

They are frequently used in the invariant theory of

n x n matrices to find the generators and relations of the ring of invariants, and therefore are useful in answering questions similar to that posed by Hilbert's fourteenth problem.

Examples

• The Cayley–Hamilton theorem says that every square matrix satisfies its own characteristic polynomial. This also implies that all square matrices satisfy $$\backslash operatorname\backslash left(A^n\backslash right)\; -\; c\_\; \backslash operatorname\backslash left(A^\backslash right)\; +\; \backslash cdots\; +\; (-1)^n\; n\; \backslash det(A)\; =\; 0\backslash ,$$ where the coefficients

c _{i} are given by the elementary symmetric polynomials of the eigenvalues of .

• All square matrices satisfy $$\backslash operatorname(A)\; =\; \backslash operatorname\backslash left(A^\backslash mathsf\backslash right).\backslash ,$$

References

.