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

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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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Tensor unfoldings preserve the Frobenius inner product

Statement

For tensors $A, B$ of the same finite shape, every mode unfolding and every cut unfolding preserves the Frobenius inner product and norm.

Facts & Assumptions

Given: A field $F \in {R, C}$ , an order $d \geq 2$ , and tensors $A, B \in F^{n_{1} \times \dots \times n_{d}}$ .

[F1]

Finite tensor arrays and the Frobenius inner product: each unfolding uses a fixed lexicographic ordering of the same tensor indices, and the Frobenius inner product sums conjugate-entry times entry over all indices.

Proof

1.1givenF1construct

For a mode- $k$ unfolding, the map from a multi-index $(i_{1}, \dots, i_{d})$ to the matrix position consisting of row $i_{k}$ and the ordered tuple of all other indices is a bijection: the row and column tuple recover every original index exactly once. Thus the entrywise sum defining $⟨ A_{(k)}, B_{(k)} ⟩_{F}$ is precisely the same finite sum as $⟨ A, B ⟩_{F}$ .

2.1givenF1algebra∎

For a cut- $k$ unfolding, splitting the tuple into $(i_{1}, \dots, i_{k})$ and $(i_{k + 1}, \dots, i_{d})$ is also a bijection with the original multi-index set. Its entrywise inner-product sum therefore equals $⟨ A, B ⟩_{F}$ as well. Taking $B = A$ and nonnegative square roots proves each norm identity.

Depends on

Finite tensor arrays and the Frobenius inner product

Used by

Exact TT ranks and root-sum-square truncation error Theorem
HOSVD quasioptimality for multilinear rank Theorem

Dependency tree · one level

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Sources

Tamara G. Kolda and Brett W. Bader, Tensor Decompositions and Applications, SIAM Review 51(3) (2009), §2.4 (standard reference, not scraped)
Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle, A Multilinear Singular Value Decomposition, SIAM Journal on Matrix Analysis and Applications 21(4) (2000), §2 (standard reference, not scraped)