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 $j$th row and $i$th column, for all indices i and j:

$a_{i,j}=a_{j,i}^{*}.$

If the conjugate transpose of a matrix $A$ is denoted by $A^{\dagger }$, then this can concisely be written as

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

For example,

${\begin{bmatrix}3&2+i\\2-i&1\end{bmatrix}}$

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