---
tag: 06J4PXRG22
kind: statement
role: theorem
name: Spectral theorem
msc: [47B15, 15A18]
uses: [0MB8HQW495, 0FWJ0VW561, 0B5H4208Z9]
sources:
  - url: https://en.wikipedia.org/wiki/Spectral_theorem
    locator: Hermitian maps and Hermitian matrices
same_as:
  wikidata: Q1425077
  wikipedia: Spectral theorem
---

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be a [[self-adjoint|0NEGN5CRSC]] linear map.
Then $V$ has an [[orthonormal basis|0S9GGTR0EX]] that consists of [[eigenvectors|0N6960W0BV]] of $T$.
The corresponding eigenvalues are real.

## Remarks

In this basis, the matrix of $T$ is a real diagonal matrix.
Equivalently, $V$ is the orthogonal direct sum of the eigenspaces of $T$.
Over $\C$, the same conclusion holds, with complex eigenvalues, for each [[normal operator|0C8BR8XRTR]].
The matrix form is the [[spectral theorem for Hermitian matrices|0DGZBQWZWN]].

## Explanations

### L1 human

A self-adjoint map is a pure stretch along perpendicular axes.
The axes are the eigenvectors, and the stretch factors are the real eigenvalues.
Every symmetric real matrix is diagonal in a suitable orthonormal basis.