Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-28
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.

FALSE: The operator norm is always the largest modulus of an eigenvalue

Statement

The operator norm is always the largest modulus of an eigenvalue.

Facts & Assumptions

Given: The nilpotent matrix N=(0200) on R2.

[L1]

The operator norm equals the largest singular value (The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector).

Refutation

technique · direct
1.1algebra

The characteristic polynomial of N is x2, so every eigenvalue of N has modulus 0.

2.1L1step 1.1algebra∎

For a unit vector (x,y)T, one has ∥N(x,y)T∥=2∣y∣≤2, with equality at (0,1)T. Thus ∥N∥=2, which by [L1] is also its largest singular value. Because 2≠0, the operator norm is not the largest eigenvalue modulus.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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