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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
- Let K be a field, p≥1, and A∈ Mₚ(K). If χ_A(t)=∏_i<p(t-λᵢ) in K[t], then the transfer-matrix trace series is ∑_i<p(1-λᵢ x)⁻¹ Corollary
- Under End_F(V)≅ V^*⊗_FV, tensor contraction is the trace Corollary
- Lefschetz number of a finite CW self-map Definition
- Principal curvatures, Gaussian curvature, and mean curvature of an oriented hypersurface Definition
- Ricci curvature Definition
- The character χ_V(g)=tr(ρ_V(g)) of a finite-dimensional complex representation Definition
- The norm N_K/F and trace Tr_K/F of a finite field extension Definition
- Trace form of a representation Definition
- Descent of the two nontrivial characters of C₃ Example
- The categorical trace of a linear endomorphism is its matrix trace Example
- The identity endomorphism and its coordinate-independent trace Example
- The rational simple quaternion block Example
- Hopf trace formula Lemma
- The trace of an idempotent is its rank as a field scalar Lemma
- Field norm and trace agree with the determinant and trace of multiplication by an element Theorem
- If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then tr(T)=∑_i<nλᵢ: trace is the sum of the eigenvalues counted with algebraic multiplicity Theorem
- Norm is multiplicative, trace is F-linear, and both are transitive in towers Theorem
Dependency tree · two levels
28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- S. Axler, Linear Algebra Done Right, 4th ed., Definition 8.51 (standard reference, not scraped)