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The basis-independent trace of an endomorphism of a finite-dimensional vector space
Definition
Let be a finite-dimensional vector space over and let be linear. Choose any ordered basis of and define the trace of the endomorphism by
This is independent of the chosen basis: matrices of the same endomorphism in two ordered bases are similar, and similar matrices have equal trace. When is the zero space, the empty basis gives trace .
Depends on
- Finite-dimensional vector space, and its dimension $\dim_F V$; infinite-dimensional means having no finite basis
- Coordinate columns $[v]_{\mathcal B}$ and matrices $[T]_{\mathcal B}^{\mathcal C}$ of linear maps relative to ordered bases
- Similar matrices have the same trace
- Similarity is an equivalence relation, and two matrices represent the same endomorphism in two bases exactly when they are similar
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
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Sources
- S. Axler, Linear Algebra Done Right, 4th ed., Definition 8.51 (standard reference, not scraped)