In mathematics, a trace identity is any equation involving the trace of a matrix.
Properties
editTrace identities are invariant under simultaneous conjugation.
Uses
editThey are frequently used in the invariant theory of 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
edit- The Cayley–Hamilton theorem says that every square matrix satisfies its own characteristic polynomial. This also implies that all square matrices satisfy where the coefficients are given by the elementary symmetric polynomials of the eigenvalues of A.
- All square matrices satisfy
See also
edit- Trace inequality – inequalities involving linear operators on Hilbert spaces
References
editRowen, Louis Halle (2008), Graduate Algebra: Noncommutative View, Graduate Studies in Mathematics, vol. 2, American Mathematical Society, p. 412, ISBN 9780821841532.