Normal matrix/2x2/Explicit/Example

In the two-dimensional situation, the condition for normality is

${\displaystyle {}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}{\begin{pmatrix}{\overline {a}}&{\overline {c}}\\{\overline {b}}&{\overline {d}}\end{pmatrix}}={\begin{pmatrix}{\overline {a}}&{\overline {c}}\\{\overline {b}}&{\overline {d}}\end{pmatrix}}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\,;}$

looking at the entries, this translates to the conditions

${\displaystyle {}b{\overline {b}}=c{\overline {c}}\,}$

and

${\displaystyle {}a{\overline {c}}+b{\overline {d}}={\overline {a}}b+{\overline {c}}d\,.}$

Besides diagonal matrices and rotation matrices, also real matrices of the form

${\displaystyle {\begin{pmatrix}a&b\\-b&a\end{pmatrix}}}$

have this property.