Trace identity

Trace identity

In mathematics, a trace identity is any equation involving the trace of a matrix. For example, the Cayley-Hamilton theorem says that every matrix satisfies its own characteristic polynomial.

Trace identities are invariant under simultaneous conjugation. 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 matrices satisfy::{ m tr}(A^n)-{ m tr} (A){ m tr}(A^{n-1})+cdots+(-1)^n 2det(A)=0.,
* Symmetric matrices satisfy::{ m tr}(A)={ m tr}(A^T).,

See also

* Invariant polynomial


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