LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Trace of a product of Hermitian matrices is real

Statement

For Hermitian matrices $A, B \in C^{n \times n}$ , $n \geq 0$ , the trace $tr (A B)$ is real.

Facts & Assumptions

Given: $A^{*} = A$ and $B^{*} = B$ in an orthonormal coordinate basis; trace is the diagonal sum, including the empty sum when $n = 0$ .

[F1]

Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space: self-adjoint means $T^{*} = T$ .

[F2]

In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix: the matrix of an adjoint in orthonormal bases is the conjugate transpose.

[F3]

Adjoints satisfy $(S + T)^{*} = S^{*} + T^{*}$ , $(λ T)^{*} = \overline{λ} T^{*}$ , $(S T)^{*} = T^{*} S^{*}$ , and $T^{* *} = T$ : $(S T)^{*} = T^{*} S^{*}$ for compatible finite-dimensional maps.

[F4]

For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ : for compatible rectangular matrices, $tr (U V) = tr (V U)$ .

[F5]

The trace $tr (A)$ as the sum of the diagonal entries: trace is the sum of the diagonal entries, and the empty diagonal sum is zero.

Proof

technique · direct

1.1givenF2F5algebra

Conjugating the diagonal sum in [F5] entry by entry and using [F2] gives $\overline{tr (A B)} = tr ((A B)^{*})$ . For $n = 0$ , both sides are the empty sum $0$ .

2.1givenF1F3F4step 1.1algebra∎

By [F1] and [F3], $(A B)^{*} = B^{*} A^{*} = B A$ . Then [F4] gives $tr (B A) = tr (A B)$ , so step 1.1 says $tr (A B)$ equals its conjugate and is real.

Depends on

Used by

Finite semidefinite primal-dual pair Definition
Matrix multiplicative-weights game and update Definition
Parameterized variational quantum objective Definition
Variational eigenvalue objective Definition
Flagged trace ratio returns a Gibbs-state observable mean Lemma
Normalized Hermitian inequality SDPs have equal primal and dual values Lemma
Normalized Hermitian observable defines a two-outcome POVM Lemma
Matrix multiplicative-weights regret bound Theorem

Dependency tree · two levels

17 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

Joran van Apeldoorn et al., Quantum SDP-solvers: Better upper and lower bounds (standard reference, not scraped)