The IsOrthogonal(A) command determines if $A$ is an orthogonal Matrix ($A.A^{+}=\mathrm{Id}$, where $A^{+}$ is the transpose and $\mathrm{Id}$ is the identity Matrix).

In general, the IsOrthogonal command returns true if it can determine that Matrix $A$ is orthogonal, false if it can determine that the Matrix is not orthogonal, and FAIL otherwise.

The IsUnitary(A) command determines if $A$ is a unitary Matrix ($A.A^{*}=\mathrm{Id}$, where $A^{*}$ is the Hermitian transpose and $\mathrm{Id}$ is the identity Matrix).

In general, the IsUnitary command returns true if it can determine that Matrix $A$ is unitary, false if it can determine that the Matrix is not unitary, and FAIL otherwise.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$G≔\mathrm{RotationMatrix}\left(\frac{\mathrm{\pi }}{7}\right)$

$G≔\left[\begin{array}{cc}\cos \left(\frac{\mathrm{\pi }}{7}\right)& -\sin \left(\frac{\mathrm{\pi }}{7}\right)\\ \sin \left(\frac{\mathrm{\pi }}{7}\right)& \cos \left(\frac{\mathrm{\pi }}{7}\right)\end{array}\right]$

$\mathrm{IsOrthogonal}\left(G\right)$

$\mathrm{true}$

$\mathrm{map}\left(\mathrm{simplify}\,G·G^{\mathrm{\%T}}\right)$

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$Q≔⟨⟨\frac{\mathrm{sqrt}\left(10\right)\cdot 3}{10}\,-\frac{\mathrm{sqrt}\left(10\right)}{10}⟩|⟨\frac{\mathrm{sqrt}\left(10\right)\mathrm{I}}{10}\,\frac{3\mathrm{sqrt}\left(10\right)\mathrm{I}}{10}⟩⟩$

$Q≔\left[\begin{array}{cc}\frac{3\sqrt{10}}{10}& \frac{\mathrm{I}}{10}\sqrt{10}\\ -\frac{\sqrt{10}}{10}& \frac{3\mathrm{I}}{10}\sqrt{10}\end{array}\right]$

$\mathrm{IsOrthogonal}\left(Q\right)$

$\mathrm{false}$

$\mathrm{IsUnitary}\left(Q\right)$

$\mathrm{true}$