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
\( {\displaystyle \operatorname {tr} \left(A^{n}\right)-\operatorname {tr} (A)\operatorname {tr} \left(A^{n-1}\right)+\cdots +(-1)^{n}n\det(A)=0.\,} \)
All square matrices satisfy
\( {\displaystyle \operatorname {tr} (A)=\operatorname {tr} \left(A^{\mathsf {T}}\right).\,} \)
References
Rowen, Louis Halle (2008), Graduate Algebra: Noncommutative View, Graduate Studies in Mathematics, 2, American Mathematical Society, p. 412, ISBN 9780821841532.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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