Hermitian Transpose

The Hermitian transpose of a matrix is the transpose together with the replacement of each entry by its complex conjugate. It is also referred to as the adjoint or Hermitian adjoint of a matrix (not to be confused with the classical adjoint or adjugate, discussed elsewhere in this chapter.)

$\blacktriangleright$ To compute the Hermitian transpose of a matrix

1.

Place the insertion point in the matrix.

2.

From the Matrices submenu, choose Hermitian Transpose.

$\left(\vphantom{ \begin{array}{cc} a+ib & c+id \\ e+if & g+ih \end{array} }\right.$$\begin{array}{cc} a+ib & c+id \\ e+if & g+ih \end{array}$$\left.\vphantom{ \begin{array}{cc} a+ib & c+id \\ e+if & g+ih \end{array} }\right)$, Hermitian transpose: $\left(\vphantom{ \begin{array}{cc} a-ib & e-if \\ c-id & g-ih \end{array} }\right.$$\begin{array}{cc} a-ib & e-if \\ c-id & g-ih \end{array}$$\left.\vphantom{ \begin{array}{cc} a-ib & e-if \\ c-id & g-ih \end{array} }\right)$

$\left(\vphantom{ \begin{array}{cc} 2+i & -i \\ 4-i & 2+i \end{array} }\right.$$\begin{array}{cc} 2+i & -i \\ 4-i & 2+i \end{array}$$\left.\vphantom{ \begin{array}{cc} 2+i & -i \\ 4-i & 2+i \end{array} }\right)$, Hermitian transpose: $\left(\vphantom{ \begin{array}{cc} 2-i & 4+i \\ i & 2-i \end{array} }\right.$$\begin{array}{cc} 2-i & 4+i \\ i & 2-i \end{array}$$\left.\vphantom{ \begin{array}{cc} 2-i & 4+i \\ i & 2-i \end{array} }\right)$

You can also get the Hermitian transpose of a matrix by using the superscript H.

$\blacktriangleright$ Evaluate

$\left(\vphantom{ \begin{array}{rr} i & 2+i \\ 4i & 3-2i \end{array} }\right.$$\begin{array}{rr} i & 2+i \\ 4i & 3-2i \end{array}$$\left.\vphantom{ \begin{array}{rr} i & 2+i \\ 4i & 3-2i \end{array} }\right)^{{H}}_{}$ =  $\left(\vphantom{ \begin{array}{cc} -i & -4i \\ 2-i & 3+2i \end{array} }\right.$$\begin{array}{cc} -i & -4i \\ 2-i & 3+2i \end{array}$$\left.\vphantom{ \begin{array}{cc} -i & -4i \\ 2-i & 3+2i \end{array} }\right)$