Trace identity

Trace identities are invariant under simultaneous conjugation.

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

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

• {{annotated link|Trace inequality}}

{{reflist}}

{{citation|title=Graduate Algebra: Noncommutative View|volume=2|series=Graduate Studies in Mathematics|first=Louis Halle|last=Rowen|publisher=American Mathematical Society|year=2008|isbn=9780821841532|page=412|url=https://books.google.com/books?id=8svFC09gGeMC&pg=PA412}}.

Category:Invariant theory

Category:Linear algebra