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ZixuanZhang
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Eigenvalues of Hermitian matrices

Every eigenvalue of a Hermitian matrix is real, eigenvectors for distinct eigenvalues are orthogonal, and a symmetric matrix admits a real eigenvector for each eigenvalue.

Theorem 4.16

For a Hermitian matrix of size ,

  1. Every eigenvalue of is real.

  2. Eigenvectors corresponding to distinct eigenvalues are orthogonal.

  3. If is symmetric, then for each eigenvalue , we can choose a real eigenvector so that (2) becomes

Proof

  1. For an eigenvector with eigenvalue , Hermiticity gives , hence . Since , and .

  2. Let be eigenvectors with eigenvalues . Then , where part (1) makes both eigenvalues real. Since , .

  3. For real symmetric with real , write with . Then and separately. At least one of is non-zero because is an eigenvector, and that vector is a real eigenvector.

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