ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

A real Schur block represents a conjugate eigenvalue pair

Example

For real $a$ and $b > 0$ , let

$T = (\begin{array}{cc} a & b \\ - b & a \end{array}), Q = I .$

Then $(Q, T)$ is a real Schur pair and the two eigenvalues are $a + i b$ and $a - i b$ .

Facts & Assumptions

Given: Real $a$ , positive $b$ , and the matrices $T, Q$ above.

[F1]

Real and complex Schur forms: for real $A$ , a real Schur pair $(Q, T)$ satisfies $Q^{T} Q = I$ and $A = Q T Q^{T}$ , with $T$ block upper triangular and each diagonal block either real $1 \times 1$ or real $2 \times 2$ with negative characteristic-polynomial discriminant.

[F2]

The real Francis double-shift polynomial is real and annihilates its trailing block: the characteristic polynomial of a real $2 \times 2$ block is determined by its trace and determinant.

Verification

1.1givenF1algebra

Since $Q = I$ , $Q^{T} Q = I$ and $Q T Q^{T} = T$ . The only diagonal block is $T$ , and its characteristic polynomial is $(λ - a)^{2} + b^{2}$ , with discriminant $- 4 b^{2} < 0$ . Thus [F1] says $(Q, T)$ is a real Schur pair.

2.1givenF2algebra∎

The roots of $(λ - a)^{2} + b^{2}$ are $λ = a \pm i b$ . Equivalently, [F2] gives trace $2 a$ and determinant $a^{2} + b^{2}$ , yielding the same conjugate roots.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

James W. Demmel, Applied Numerical Linear Algebra, §4.4.8 (standard reference, not scraped)
David Bindel, CS 6210 Matrix Computations, Hessenberg QR iteration (standard reference, not scraped)