ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Courant-Fischer is checked on an explicit 3x3 symmetric matrix

Example

For the symmetric matrix

$D = diag (4, 2, 1),$

the Rayleigh quotient is

$R_{D} (x, y, z) = \frac{4 x^{2} + 2 y^{2} + z^{2}}{x^{2} + y^{2} + z^{2}},$

so the Courant-Fischer formulas recover the ordered eigenvalues $4, 2, 1$ .

Facts & Assumptions

Given: The diagonal matrix $D = diag (4, 2, 1)$ on $R^{3}$ with the standard inner product.

[L1]

Courant-Fischer characterises the ordered eigenvalues of a real self-adjoint operator by min-max formulas (Courant-Fischer min-max principle for self-adjoint endomorphisms on finite-dimensional real inner product spaces).

Verification

technique · direct

1.1

For every nonzero $(x, y, z)$ , the displayed formula shows $1 \leq R_{D} (x, y, z) \leq 4$ , with the value $4$ at $e_{1}$ , the value $2$ at $e_{2}$ , and the value $1$ at $e_{3}$ . On $span (e_{2}, e_{3})$ the quotient is at most $2$ , and on $span (e_{1}, e_{2})$ it is at least $2$ .

algebra

2.1

Therefore $\max_{v \neq 0} R_{D} (v) = 4$ , $\min_{v \neq 0} R_{D} (v) = 1$ , and the $k = 2$ Courant-Fischer min-max and max-min values are both $2$ . This matches the ordered eigenvalues $4, 2, 1$ , exactly as [L1] predicts.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

MIT 18.409, Lecture 3: Courant-Fischer and Rayleigh quotients (standard reference, not scraped)