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Section 3.5 Hermitian Adjoint

The Hermitian adjoint of a matrix is the same as its transpose except that along with switching row and column elements you also complex conjugate all the elements. If all the elements of a matrix are real, its Hermitian adjoint and transpose are the same. In terms of components,

\begin{equation} \left(A_{ij}\right)^\dagger=A_{ji}^*\text{.}\tag{3.5.1} \end{equation}

For example, if

\begin{equation} A = \begin{pmatrix} 1\\ i\\ -2i \end{pmatrix}\tag{3.5.2} \end{equation}


\begin{equation} A^\dagger = \begin{pmatrix} 1\amp -i\amp 2i \end{pmatrix}\text{.}\tag{3.5.3} \end{equation}

A matrix is called Hermitian if it is equal to its adjoint, \(A = A^\dagger\text{.}\)

Compute \(B^\dagger\) if

\begin{equation} B=\begin{pmatrix} 1\amp i\\ -5i\amp i \end{pmatrix}\text{.}\tag{3.5.4} \end{equation}
\begin{equation} B^\dagger = (B^T)^* = \begin{pmatrix}1\amp -5i\\i\amp i\end{pmatrix}^* = \begin{pmatrix}1\amp 5i\\-i\amp -i\end{pmatrix}\text{.}\tag{3.5.5} \end{equation}