Trace identity

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

Properties

Trace identities are invariant under simultaneous conjugation.

Uses

They are frequently used in the invariant theory of $n\times n$ 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

• The Cayley–Hamilton theorem says that every square matrix satisfies its own characteristic polynomial. This also implies that all square matrices satisfy $$\mathrm{tr}\left(A^n\right)-c_{n-1}\mathrm{tr}\left(A^{n-1}\right)+\cdots +(-1{)}^{n}n\det (A)=0$$ where the coefficients $c_i$ are given by the elementary symmetric polynomials of the eigenvalues of A.
• All square matrices satisfy $$\mathrm{tr}(A)=\mathrm{tr}\left(A^{𝖳}\right).$$

See also

• Trace inequality – Concept in Hlibert spaces mathematics

References

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