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    Eigenvalues of Hermitian matrices are real, and eigenvalues of unitary matrices have modulus $1$

    P-ICV5F

    Area
    Algebra
    Topics
    Eigenvalues and Eigenvectors, Inner Product Spaces, Matrices
    Status
    draft

    Prove that the eigenvalues of a Hermitian matrix are real and those of a unitary matrix are unitary.

    Solution

    1. Every eigenvalue of a Hermitian matrix is real.

    Let \(A=A^*\) and let \(Av=\lambda v\) with \(v\ne0\). Then \[ \lambda\langle v,v\rangle =\langle Av,v\rangle =\langle v,A^*v\rangle =\langle v,Av\rangle =\overline\lambda\langle v,v\rangle. \] Since \(\langle v,v\rangle>0\), one gets \(\lambda=\overline\lambda\), so \(\lambda\in\RR\).

    2. Every eigenvalue of a unitary matrix has modulus \(1\).

    Let \(U^*U=I\) and \(Uv=\lambda v\) with \(v\ne0\). Unitary matrices preserve norms, so \[ \|v\|=\|Uv\|=\|\lambda v\|=|\lambda|\,\|v\|. \] Thus \(|\lambda|=1\).

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