ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-16

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Frobenius inner product on a real or complex matrix space

Example

On $M_{m \times n} (F)$ , where $F = R$ or $C$ , the formula

$⟨ A, B ⟩_{F} = \sum_{i < m} \sum_{j < n} A_{i j} \overline{B_{i j}}$

defines the Frobenius inner product, and

$∥ A ∥_{F}^{2} = \sum_{i < m} \sum_{j < n} ∣ A_{i j} ∣^{2} .$

Facts & Assumptions

Given: Matrices $A, B$ of one fixed $m \times n$ shape.

[L1]

Matrices of a fixed shape form a vector space under entrywise operations (The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations).

[L2]

An inner product must be linear in its first argument, conjugate symmetric, positive, and definite (Real and complex inner product spaces, with the inner product linear in the first argument).

[L3]

Complex conjugation distributes over finite sums and satisfies $z \overline{z} = ∣ z ∣^{2}$ , vanishing exactly at $z = 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[L4]

Empty finite sums have additive value zero (The product $g_{0} g_{1} \dots g_{n - 1}$ of a finite list in a monoid, by recursion, with the empty product ( $n = 0$ ) equal to the identity).

Verification

technique · direct

1.1L1L3

Entrywise operations and finite-sum algebra give linearity in $A$ . Applying conjugation termwise and using [L3] gives conjugate symmetry.

1.2L1L2L3

On the diagonal, [L3] gives the displayed sum of squared moduli. It is nonnegative and vanishes exactly when every entry vanishes, which by [L1] means $A = 0$ . Hence [L2] is satisfied.

2.1L1L4∎

If $m = 0$ or $n = 0$ , the matrix space contains only its zero matrix and [L4] makes the formula zero, so definiteness remains valid.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

Sergei Treil, Linear Algebra Done Wrong, Example 5.1.3 (standard reference, not scraped)