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\)
Section 5.2 Definition of Hermitian Matrices
Definition 5.2 . Hermitian Matrices.
On an
\(n\times m\) matrix,
\(N\text{,}\) you can take the
Hermitian adjoint (usually denoted with a dagger,
\(\dagger\) ) which means take both the (complex) conjugate
Section 2.3 and the transpose
Section 3.4 , in either order
\begin{equation}
N^\dagger=N^*{}^T\text{.}\tag{5.2.1}
\end{equation}
Two uses of the word “Hermitian”. You should not confuse the two uses of the word Hermitian . One describes an operation that you can do to matrices–to take the Hermitian adjoint (an action), the other describes a matrix with a particular form–a Hermitian matrix (an object).
An
\(n\times n\) (square) matrix
\(M\) is
Hermitian if it equals its conjugate transpose, that is, if
\begin{equation}
M^\dagger = M\text{.}\tag{5.2.2}
\end{equation}
If a matrix \(M\) is both Hermitian and real, then \(M\) is called a symmetric matrix . An anti-Hermitian matrix is one for which the Hermitian adjoint is the negative of the matrix:
\begin{equation}
M^\dagger = -M\text{.}\tag{5.2.3}
\end{equation}
A matrix which is both anti-Hermitian and real is called antisymmetric .
Identifying Hermitian Matrices.
For example, let \(M\) be a \(2\times2\) complex matrix, so that
\begin{equation}
M = \begin{pmatrix} a\amp b\\ c\amp d \end{pmatrix}\text{,}\tag{5.2.4}
\end{equation}
with \(a,b,c,d\in\CC\text{,}\) and its Hermitian adjoint is
\begin{equation}
M^\dagger = \begin{pmatrix} a^*\amp c^*\\ b^*\amp d^* \end{pmatrix}\text{,}\tag{5.2.5}
\end{equation}
If \(M\) is Hermitian, then \(M^\dagger=M\text{,}\) so, we must have
\begin{equation}
a^* = a, b^* = c, d^* = d\text{,}\tag{5.2.6}
\end{equation}
i.e. \(a\) and \(d\) are real and \(c\) is the complex conjugate of \(b\text{.}\)
In index notation, if the components of \(M\) are denoted \(m_{ij}\text{,}\) then \(M\) is Hermitian if and only if
\begin{equation}
m_{ij} = m^*_{ji}\tag{5.2.7}
\end{equation}
for all
\(i\text{,}\) \(j\text{.}\) (See also,
Section 3.5 .) Thus, the diagonal elements of a Hermitian matrix must be real, and the off-diagonal elements come in complex conjugate pairs, paired symmetrically across the main diagonal.
An Important Special Case of Hermitian Matrices. An important special case of a Hermitian matrix can be constructed from any column vector
\(v\) by computing its outer square, which in traditional vector notation would be written
\(vv^\dagger\) and in bra/ket notation would be written
\(|v\rangle\langle v|\text{.}\) (See
Section 5.6 for a description of how such operators are used to produce projections.)
