Trace identity

Equations involving the trace of a matrix

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 × 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.

Examples

  • 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).\,}

See also

  • Trace inequality – inequalities involving linear operators on Hilbert spacesPages displaying wikidata descriptions as a fallback

References

Rowen, Louis Halle (2008), Graduate Algebra: Noncommutative View, Graduate Studies in Mathematics, vol. 2, American Mathematical Society, p. 412, ISBN 9780821841532.