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
In Branch
Proved By
Source Spectral Theorem (Wikipedia)
Sources
1. Spectral Theorem (Wikipedia)
Wikimedia FoundationCyclic vectors and simple spectrum
That is, a bounded self-adjoint operator that admits a cyclic vector should be thought of as the infinite-dimensional generalization of a self-adjoint matrix with distinct eigenvalues (i.e., each eigenvalue has multiplicity one).
Proved By: Augustin-Louis Cauchy, Lead
Augustin-Louis Cauchy proved the spectral theorem for symmetric matrices, i.e., that every real, symmetric matrix is diagonalizable.
In Group: Linear Algebra and Matrix Theory, Lead section
In linear algebra and functional analysis, a spectral theorem is
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