Every symmetric (or, in the complex case, Hermitian) matrix can be diagonalized by an orthonormal basis of eigenvectors, and the analogous statement holds for self-adjoint operators on a Hilbert space. It is one of the most widely used structural results in linear algebra and functional analysis.
Facts
StatementIf A is a real symmetric matrix, or a complex Hermitian matrix, there exists an orthonormal basis of eigenvectors of A, and every eigenvalue of A is real. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Sources
1. Spectral Theorem (Wikipedia)
Wikimedia FoundationStatement section, Hermitian caseQuote, Statement section, Hermitian case
If A is Hermitian on V, then there exists an orthonormal basis of V consisting of eigenvectors of A. Each eigenvalue of A is real.
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