Trace identity

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

Example

For example, the Cayley–Hamilton theorem says that every matrix satisfies its own characteristic polynomial.

Properties

Trace identities are invariant under simultaneous conjugation.

Uses

They are frequently used in the invariant theory of n×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

• By the Cayley–Hamilton theorem, all square matrices satisfy

${\rm tr}(A^n)-{\rm tr} (A){\rm tr}(A^{n-1})+\cdots+(-1)^n \det(A)=0.\,$

• All square matrices satisfy

${\rm tr}(A)={\rm tr}(A^\text{T}).\,$

References

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