Hermitian matrices

A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex elements which is equal to its own conjugate transpose — that is, the element in the $i$th row and $j$th column is equal to the complex conjugate of the element in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} th row and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} th column, for all indices i and j:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{i,j} = a_{j,i}^*. }

If the conjugate transpose of a matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\dagger} , then this can concisely be written as

$A=A^{\dagger }.\,$

For example,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix}3&2+i\\ 2-i&1\end{bmatrix} }

All eigenvalues of a Hermitian matrix are real, and, moreover, eigenvectors with distinct eigenvalues are orthogonal. The typical example of a Hermitian matrix in physics is the Hamiltonian (specially in quantum mechanics).

References

• Hermitian matrix entry in Wikipedia