Hermitian Matrix

linear-algebra

Definition

Hermitian Matrix

Let $A$ be a square matrix over the complex numbers. The matrix is Hermitian if it is equal to its conjugate transpose, that is,

$$A=A^{\ast }$$

Equivalently, its entries satisfy

$$a_{ij}=\overline{a_{ji}}\forall i,j.$$

Spectral Representation

If $A$ is Hermitian, then it has a spectral decomposition of the form

$$A=\sum _{i\ge 0}{\lambda }_i{P}_i,$$

where ${\lambda }_i$ are real eigenvalues of $A$ and each $P_i$ is an orthogonal projection.

Note that the convergence is understood entrywise in the finite-dimensional case.

Properties

• A Hermitian matrix has real eigenvalues.
• If all entries are real, then a Hermitian matrix is the same as a symmetric matrix.