Trace identity

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 $$n \times n$$ 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 $$\operatorname{tr}\left(A^n\right) - c_{n-1} \operatorname{tr}\left(A^{n - 1}\right) + \cdots + (-1)^n n \det(A) = 0\,$$ where the coefficients $$c_i$$ are given by the elementary symmetric polynomials of the eigenvalues of $A$.
 * All square matrices satisfy $$\operatorname{tr}(A) = \operatorname{tr}\left(A^\mathsf{T}\right).\,$$