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)$