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

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The eigenvalues of the orthogonal compression of a self-adjoint endomorphism to a hyperplane interlace those of the original endomorphism

Statement

Let $T : V \to V$ be self-adjoint on an $n$ -dimensional real inner product space, let $W \subset V$ be a hyperplane, and let

$C : = P_{W} \circ T ∣_{W} : W \to W$

be the orthogonal compression, where $P_{W}$ is the orthogonal projection onto $W$ . If the eigenvalues of $T$ and $C$ are ordered by

$λ_{1} \geq \dots \geq λ_{n}, μ_{1} \geq \dots \geq μ_{n - 1},$

then

$λ_{i} \geq μ_{i} \geq λ_{i + 1} (1 \leq i \leq n - 1) .$

Facts & Assumptions

Given: A self-adjoint endomorphism $T : V \to V$ on an $n$ -dimensional real inner product space, a hyperplane $W \subset V$ , and its orthogonal projection $P_{W}$ .

[L1]

Every vector decomposes uniquely as a vector in $W$ plus a vector in $W^{⊥}$ (For a subspace $W$ of a finite-dimensional inner product space, $V = W \oplus W^{⊥}$ , The orthogonal projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$ ).

[L2]

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

Proof

technique · direct

1.1

For $u, v \in W$ , the decomposition in [L1] gives $⟨ P_{W} T u, v ⟩ = ⟨ T u, v ⟩ = ⟨ u, T v ⟩ = ⟨ u, P_{W} T v ⟩$ , so $C$ is self-adjoint on $W$ . Also $R_{C} (u) = \frac{⟨ C u, u ⟩}{⟨ u, u ⟩} = \frac{⟨ T u, u ⟩}{⟨ u, u ⟩} = R_{T} (u)$ for every nonzero $u \in W$ .

L1algebra

2.1

For any $1 \leq i \leq n - 1$ , Courant-Fischer in $W$ and step 1.1 give $μ_{i} = \max_{\dim U = i, U \subseteq W} \min_{0 \neq u \in U} R_{C} (u) = \max_{\dim U = i, U \subseteq W} \min_{0 \neq u \in U} R_{T} (u) \leq λ_{i}$ , because the maximisation is over fewer $i$ -dimensional subspaces than in [L2]. Likewise $μ_{i} = \min_{\dim L = n - i, L \subseteq W} \max_{0 \neq u \in L} R_{C} (u) = \min_{\dim L = n - i, L \subseteq W} \max_{0 \neq u \in L} R_{T} (u) \geq λ_{i + 1}$ , because $n - i = (n - 1) - i + 1$ is the subspace dimension appearing in Courant-Fischer for the $(i + 1)$ -st eigenvalue of $T$ . Hence $λ_{i} \geq μ_{i} \geq λ_{i + 1}$ .

L2step 1.1∎

Depends on

Used by

The eigenvalues of a principal 2x2 submatrix interlace those of a 3x3 symmetric matrix Example

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

Christoph Helmberg et al., An interlacing property of the signless Laplacian of threshold graphs (standard reference, not scraped)