Hermitian Matrix

linear-algebra

Definition

Hermitian Matrix

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

Equivalently, its entries satisfy

Spectral Representation

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

where are real eigenvalues of and each 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.