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Trace identity

Equations involving the trace of a matrix

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

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Properties

Trace identities are invariant under simultaneous conjugation.

They are frequently used in the invariant theory of n × n {\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.

The Cayley–Hamilton theorem says that every square matrix satisfies its own characteristic polynomial. This also implies that all square matrices satisfy tr ⁡ ( A n ) − c n − 1 tr ⁡ ( A n − 1 ) + ⋯ + ( − 1 ) n n det ( A ) = 0 {\displaystyle \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 {\displaystyle c_{i}} are given by the elementary symmetric polynomials of the eigenvalues of A.
All square matrices satisfy tr ⁡ ( A ) = tr ⁡ ( A T ) . {\displaystyle \operatorname {tr} (A)=\operatorname {tr} \left(A^{\mathsf {T}}\right).\,}