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

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.

Column-stochastic matrices preserve mass and contract ℓ1

Statement

If $P$ is a finite column-stochastic matrix, then $P$ maps column probability vectors to column probability vectors, preserves coordinate sums on all real vectors, and satisfies $∥ P z ∥_{1} \leq ∥ z ∥_{1}$ for every $z \in R^{n}$ .

Facts & Assumptions

Given: $P$ is column-stochastic in the sense of Finite column-stochastic matrices and probability vectors.

[F1]

Finite column-stochastic matrices and probability vectors: $P_{i j} \geq 0$ , each column sums to one, a probability vector has nonnegative coordinates summing to one, and $∥ z ∥_{1} = \sum_{j} ∣ z_{j} ∣$ .

Proof

Given: A finite column-stochastic matrix $P$ and a vector $z \in R^{n}$ .

Proof technique: apply the triangle inequality coordinatewise and then use each column sum.

1.1givenF1algebra

For every $z$ , finite summation and [F1] give $\sum_{i} (P z)_{i} = \sum_{j} (\sum_{i} P_{i j}) z_{j} = \sum_{j} z_{j}$ . If $z$ is a column probability vector, each coordinate of $P z$ is a sum of nonnegative terms, so $P z \geq 0$ and its coordinate sum is one.

2.1F1step 1.1algebra∎

For each row $i$ , $∣ (P z)_{i} ∣ = ∣ \sum_{j} P_{i j} z_{j} ∣ \leq \sum_{j} P_{i j} ∣ z_{j} ∣$ because $P_{i j} \geq 0$ . Summing over $i$ and regrouping finite sums gives $∥ P z ∥_{1} \leq \sum_{j} ∣ z_{j} ∣ \sum_{i} P_{i j} = ∥ z ∥_{1}$ .

Depends on

Finite column-stochastic matrices and probability vectors

Used by

Dependency tree · two levels

2 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

Ilse C. F. Ipsen and Teresa M. Selee, Ergodicity Coefficients Defined by Vector Norms (standard reference, not scraped)