For $A\in M_{m\times n}(F)$ and $B\in M_{n\times m}(F)$, $\mathrm{tr}(AB)=\mathrm{tr}(BA)$

Depends on

• The trace $\operatorname{tr}(A)$ as the sum of the diagonal entries
• Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes
• Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule
• Field

Used by

• Similar matrices have the same trace Corollary
• The only scalar multiples of a root that are roots are plus or minus the root Corollary
• Finite adiabatic path model Definition
• Finite quantum mutual, conditional, and conditional mutual information Definition
• Finite-dimensional quantum relative entropy Definition
• Normalized output law for finite quantum query branches Definition
• Parameterized variational quantum objective Definition
• The character χ_V(g)=tr(ρ_V(g)) of a finite-dimensional complex representation Definition
• A 2 by 3 matrix and a 3 by 2 matrix give square products of different sizes but equal traces Example
• Matrix-unit rotation for a k–Mₙ(k) Morita pair Example
• Qubit dephasing channel in Kraus and Choi forms Example
• Qubit dephasing decreases quantum relative entropy Example
• A finite POVM defines a classical-output quantum channel Lemma
• A finite verifier induces an effect on its witness Lemma
• A uniform positive eigenvalue floor controls log-determinant error Lemma
• Computational-basis instruments and conditional states on two registers Lemma
• Fidelity classical measurement minimum Lemma
• Finite square-root measurement packing bound for cq states Lemma
• Flagged trace ratio returns a Gibbs-state observable mean Lemma
• Global Lipschitz ODE flows and the C1 density formula Lemma
• Hopf trace formula Lemma
• Isometric invariance of finite quantum relative entropy Lemma
• Measurement law of a coherent sampler Lemma
• Normalized Hermitian observable defines a two-outcome POVM Lemma
• One-bit J-gate teleportation branch identity Lemma
• Optimal Cloner Attains Bound Lemma
• Optimal Cloning Worst Case Upper Bound Lemma
• Pauli measurement projectors and total probability Lemma
• Pauli projectors define a finite measurement instrument Lemma
• Phase-estimation finite outcome law Lemma
• Positive Marginals Imply Joint Support Lemma
• Relative modular representation of finite quantum relative entropy Lemma
• The finite classical key-copy map is a quantum channel Lemma
• The maximum trace over q-dimensional orthonormal frames is the sum of the top q eigenvalues Lemma
• The trace form detects semisimplicity over the complex numbers Lemma
• Trace of a product of Hermitian matrices is real Lemma
• Trace of a product of positive semidefinite matrices is nonnegative Lemma
• Zero-probability Pauli measurement branch Lemma
• Degree-zero Hochschild homology is bimodule coinvariants Proposition
• For a complex character, χ(1)=dim V, χ is a class function, and |χ(g)|≤χ(1) with equality exactly at scalars Proposition

…and 31 more results.

Dependency tree · two levels

10 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., Theorem 8.49 (standard reference, not scraped)