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