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Hermitian

mathematical entities are named hermitian, after the mathematician Charles Hermite.

Matrices

A hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries so that the matrix is equal to its own conjugate transpose - that is, if the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:

<math>a_{i,j} = \overline{a_{j,i}}<math>

For example,

<math>\begin{bmatrix}3&2+i\\

2-i&1\end{bmatrix}<math> is a hermitian matrix.

In case the matrix has only real entries, a matrix is Hermitian if and only if its is symmetric with respect to the (top left to bottom right) diagonal of the the matrix.

Every Hermitian matrix is normal, and the finite-dimensional spectral theorem applies. It says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This means that all eigenvalues of a Hermitian matrix are real, and, moreover, eigenvectors with distinct eigenvalues are orthogonal. It is possible to find an orthonormal basis of Cⁿ consisting only of eigenvectors.

If the eigenvalues of a Hermitian matrix are all positive, then the matrix is positive definite. Matrix theorists sometimes refer to real Hermitian matrices as symmetric matrices, since indeed they are symmetric with respect to the diagonal.

Elements of a C*-algebra

In a C*-algebra, we generalise this notion as discussed in self-adjoint.

See also

For related concepts including symmetric and self-adjoint unbounded operators see Self-adjoint operator.