DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Invariant subspaces, restrictions, and induced quotient operators

Definition

Let $T : V \to V$ be an endomorphism. A linear subspace $W \leq V$ is $T$ -invariant when $T (W) \subseteq W$ . For such a $W$ , the restriction of $T$ to $W$ is $T ∣_{W} : W ⟶ W, w ⟼ T (w) .$ The operator induced by $T$ on the quotient is the map $\overset{ˉ}{T} : V / W ⟶ V / W, \overset{ˉ}{T} (v + W) : = T (v) + W,$ where $V / W$ is the quotient of The quotient vector space $V / W$ and its canonical projection. Its well-definedness, linearity, and relation to the canonical projection are proved in Invariance makes the induced quotient operator well defined and linear, with $π T = \overset{ˉ}{T} π$ ↗.

Depends on

Used by

An invariant subspace need not reduce an operator Counterexample
Algebraic multiplicity of a nonzero compact-operator eigenvalue Definition
Cyclic subspaces, cyclic vectors, and vector annihilators Definition
Subspace iteration and the dominant invariant subspace of a matrix Definition
Local separable trace-class determinant construction Lemma
Trace decomposition through generalized eigenspaces and the invariant quotient Lemma
Complete invariant flags are equivalent to upper-triangular matrices Proposition
For invariant W, χ_T=χ_T|_Wχ_T̄ Proposition
Invariance makes the induced quotient operator well defined and linear, with π T=T̄π Proposition
Polynomial evaluation commutes with restriction and invariant quotients Proposition

Dependency tree · two levels

5 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

K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Section 6.4 (standard reference, not scraped)
Cornell Math 4330, Quotient Spaces, Exercise QuoSpace 7 (standard reference, not scraped)