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 ${\displaystyle n\times n}$ 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
  $${\displaystyle \operatorname {tr} \left(A^{n}\right)-c_{n-1}\operatorname {tr} (A)\operatorname {tr} \left(A^{n-1}\right)+\cdots +(-1)^{n}n\det(A)=0\,}$$

  
  where the coefficients ${\displaystyle c_{i))$ are given by the elementary symmetric polynomials of the eigenvalues of A.
• All square matrices satisfy
  $${\displaystyle \operatorname {tr} (A)=\operatorname {tr} \left(A^{\mathsf {T))\right).\,}$$