# Hermitian matrices

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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).