Diagonalisation of Hermitian matrices
with unitary for Hermitian ; for real symmetric , may be taken orthogonal.
Let be a hermitian matrix of size . Then, is diagonalisable.
More specifically,
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There exists a basis of eigenvectors with
for eigenvalues ;
and equivalently,
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There exists an invertible matrix such that
with the columns of representing the eigenvectors .
Unitary and orthogonal forms
The eigenvectors can be chosen orthonormal, . Equivalently the matrix of eigenvectors can be chosen unitary, , so
For a real symmetric matrix the eigenvectors can be taken in with , equivalently orthogonal with and .
Orthonormal construction
Given a linearly independent set of vectors in , the Gram-Schmidt process replaces them step by step, subtracting multiples of earlier directions, so that each intermediate set spans the same subspace while the processed vectors become mutually orthonormal.
Finding an orthonormal basis of each eigenspace of a Hermitian matrix and taking over the distinct eigenvalues therefore yields an orthonormal set of consisting of eigenvectors of .
Related
Stated in
- Theorem 4.17§4.3.6 Unitary and Orthogonal Diagonolisation