Existing in Mathlib · Linear algebra and analysis

Finite-Dimensional Spectral Theorem

A self-adjoint linear operator may look coupled in the original space, but its mutually orthogonal eigenspaces provide coordinates in which every component is simply multiplied by its own eigenvalue.

self-adjoint operators
eigenspaces
orthogonal decomposition
diagonalization
finite-dimensional inner-product spaces

Exact theorem

Exact Mathlib statement

theorem LinearMap.IsSymmetric.diagonalization_apply_self_apply {𝕜 : Type*} [RCLike 𝕜] {E : Type*} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {T : E →ₗ[𝕜] E} [FiniteDimensional 𝕜 E] (hT : T.IsSymmetric) (v : E) (μ : Eigenvalues T) : hT.diagonalization (T v) μ = (μ : 𝕜) • hT.diagonalization v μ

The theorem at a glance

Finite-Dimensional Spectral Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory spectral-theorem poster connects self-adjointness, orthogonal eigenspaces, an isometric direct-sum coordinate map, and independent multiplication of each component by its eigenvalue. — A finite-dimensional self-adjoint operator acts diagonally in the orthogonal direct sum of its eigenspaces.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A coupled self-adjoint transformation enters from the left, separates through mutually orthogonal eigenspace channels, and exits as uncoupled coordinate scalings. — The eigenspace decomposition turns a self-adjoint operator into independent eigenvalue scalings.

Why it matters

A mathematical landmark

The spectral theorem turns self-adjoint linear operators into diagonal data and underlies large parts of linear algebra, analysis, geometry, and mathematical physics. This declaration records the exact coordinatewise action of Mathlib's finite-dimensional eigenspace decomposition.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's coordinatewise diagonalization theorem for a self-adjoint linear endomorphism of a finite-dimensional inner-product space over an RCLike field. It uses Mathlib's isometric equivalence with the direct sum of eigenspaces. It is not the infinite-dimensional spectral theorem, a theorem for arbitrary normal operators, or a claim about a user-chosen matrix basis.

Proof Atlas did not originate the spectral theorem or Mathlib's declaration.
The selected declaration is the finite-dimensional self-adjoint diagonal-action form; it does not state an infinite-dimensional projection-valued-measure theorem.
The generated explanation and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: LinearMap.IsSymmetric.diagonalization_apply_self_apply in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.finite-dimensional-spectral-theorem.v001
Source: Open the pinned upstream reference