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{{Orphan|date=December 2012}}

In [[mathematics]], a '''trace identity''' is any [[equation]] involving the [[trace (linear algebra)|trace]] of a [[matrix (mathematics)|matrix]].{{citation needed|date=March 2012}} For example, the [[Cayley–Hamilton theorem]] says that every matrix satisfies its own [[characteristic polynomial]].

Trace identities are invariant under simultaneous [[Conjugation (group theory)|conjugation]]. They are frequently used in the [[invariant theory]] of ''n''×''n'' matrices to find the [[generating set|generators]] and [[Relation (mathematics)|relations]] of the [[invariant theory|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 matrices satisfy
::<math>{\rm tr}(A^n)-{\rm tr} (A){\rm tr}(A^{n-1})+\cdots+(-1)^n \det(A)=0.\,</math>
* All matrices satisfy
::<math>{\rm tr}(A)={\rm tr}(A^T).\,</math>

== See also ==
* [[Invariant polynomial]]

[[Category:Linear algebra]]
[[Category:Invariant theory]]

{{Linear-algebra-stub}}

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