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
$\operatorname {tr} \left(A^{n}\right)-\operatorname {tr} (A)\operatorname {tr} \left(A^{n-1}\right)+\cdots +(-1)^{n}n\det(A)=0.\,$
All square matrices satisfy
$\operatorname {tr} (A)=\operatorname {tr} \left(A^{\mathsf {T}}\right).\,$

gollark: Oh, another one was PS#2DAA86DC. That was when you could run a privileged function in a coroutine and... also feed it fake events.

gollark: <@263493613860814848> How DARE YOU.

gollark: No, 'twas self-starred.

gollark: Another one was that for no apparent reason `getfenv` would sometimes return out of sandbox stuff despite it being explicitly programmed to prevent this.

gollark: Then, when I patched that, it turned out that you could also grab the coroutine directly from some internal process manager tables and feed events in a similar way.

References

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

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