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15.8: Hermitian Matrices

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  • A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. In other words, \(a_{ij}=a_{ji}^*\) for all entries. The elements in the diagonal need to be real, because these entries need to equal their complex conjugates: \(a_{ii}=a_{ii}^*\):

    \[\begin{pmatrix} a&{\color{red}b+ci}&{\color{blue}d+ei}\\ {\color{red}b-ci}&f&{\color{OliveGreen}g+hi}\\ {\color{blue}d-ei}&{\color{OliveGreen}g-hi}&j \end{pmatrix} \nonumber\]

    where all the symbols in this matrix except for \(i\) represent real numbers.

    Hermitian matrices are a generalization of the symmetric real matrices we just talked about, and they also have real eigenvalues, and eigenvectors that form a mutually orthogonal set.

    Need help? The link below contains solved examples: