Trace identity
From HandWiki
Short description: Equations involving the trace of a matrix
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 [math]\displaystyle{ n \times n }[/math] 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 [math]\displaystyle{ \operatorname{tr}\left(A^n\right) - c_{n-1}\operatorname{tr}(A) \operatorname{tr}\left(A^{n - 1}\right) + \cdots + (-1)^n n \det(A) = 0\, }[/math] where the coefficients [math]\displaystyle{ c_i }[/math] are given by the elementary symmetric polynomials of the eigenvalues of A.
- All square matrices satisfy [math]\displaystyle{ \operatorname{tr}(A) = \operatorname{tr}\left(A^\mathsf{T}\right).\, }[/math]
See also
References
Rowen, Louis Halle (2008), Graduate Algebra: Noncommutative View, Graduate Studies in Mathematics, 2, American Mathematical Society, p. 412, ISBN 9780821841532, https://books.google.com/books?id=8svFC09gGeMC&pg=PA412.
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