The basis-independent trace of an endomorphism of a finite-dimensional vector space

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
• A rank-one Hecke algebra with a perfect symmetrizing form and degenerate regular trace Counterexample
• Algebraic Lefschetz number via rational homology traces Definition
• Bloch coordinate maps Definition
• Compact-group isotypic projection Definition
• Density operator Definition
• Dual Coxeter number for the normalized invariant form Definition
• Finite adiabatic path model Definition
• Finite Hecke specialization and its regular-trace determinant Definition
• Finite instruments and conditional states Definition
• Finite POVM outcome kernel Definition
• Finite POVMs and their outcome law Definition
• Finite-dimensional quantum channel Definition
• Finite-dimensional quantum relative entropy Definition
• Finite-dimensional von Neumann entropy Definition
• Lefschetz number of a finite CW self-map Definition
• Normalized output law for finite quantum query branches Definition
• Partial trace over a finite tensor factor 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
• A projective measurement branch with probability zero Example
• Bloch coordinates of |0⟩ and H|0⟩ Example
• Descent of the two nontrivial characters of C₃ Example
• Isotypic Haar projections specialize to finite character sums Example
• Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example
• Qubit dephasing channel in Kraus and Choi forms Example
• Register endianness and a two-qubit partial trace 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
• A finite instrument with nonzero input has nonzero output space Lemma
• A finite POVM defines a classical-output quantum channel Lemma
• Acceptance-gap lower bound on final-state distance Lemma
• Additive characters are exactly one-dimensional complex representation characters Lemma
• Amplitude-amplification invariant plane Lemma
• An orthogonal finite ensemble block sum is a density operator Lemma

…and 101 more results.

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)