Hermitian Matrix

If a Matrix is Self-Adjoint, it is said to be a Hermitian matrix. Therefore, a Hermitian Matrix is defined as one for which

$${\hbox{\sf A}}= {{\hbox{\sf A}}}^\dagger,$$ (1)

where $\dagger$ denotes the Adjoint Matrix. Hermitian Matrices have Real Eigenvalues with Orthogonal Eigenvectors. For Real Matrices, Hermitian is the same as symmetrical. Any Matrix ${\hbox{\sf C}}$ which is not Hermitian can be expressed as the sum of two Hermitian matrices

$${\hbox{\sf C}} ={\textstyle{1\over 2}}({\hbox{\sf C}}+{\hbox... ...{\textstyle{1\over 2}}({\hbox{\sf C}}-{\hbox{\sf C}}^\dagger).$$ (2)

Let ${\hbox{\sf U}}$ be a Unitary Matrix and ${\hbox{\sf A}}$ be a Hermitian matrix. Then the Adjoint Matrix of a Similarity Transformation is

$$({\hbox{\sf U}}{\hbox{\sf A}}{\hbox{\sf U}}^{-1})^\dagger = [({\hbox{\sf U}}{\hbox{\sf A}})({\hbox{\sf U}}^{-1})]^\dagger = ({\hbox{\sf U}}^{-1})^\dagger({\hbox{\sf U}}{\hbox{\sf A}})^\dagger$$

$$= ({\hbox{\sf U}}^\dagger)^\dagger({\hbox{\sf A}}^\dagger{\hbox{\sf... ...sf A}}{\hbox{\sf U}}^\dagger = {\hbox{\sf U}}{\hbox{\sf A}}{\hbox{\sf U}}^{-1}.$$ (3)

The specific matrix

$${\hbox{\sf H}}(x,y,z)=\left[{\matrix{z & x+iy\cr x-iy & -z}}\right] = x{\hbox{\sf P}}_1+y{\hbox{\sf P}}_2+z{\hbox{\sf P}}_3,$$ (4)

where ${\hbox{\sf P}}_i$ are Pauli Spin Matrices, is sometimes called ``the'' Hermitian matrix.

See also Adjoint Matrix, Hermitian Operator, Pauli Spin Matrices

References

Arfken, G. ``Hermitian Matrices, Unitary Matrices.'' §4.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 209-217, 1985.