15.8: Hermitian Matrices

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: