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Student[LinearAlgebra]

 IsOrthogonal
 test if a Matrix is orthogonal
 IsUnitary
 test if a Matrix is unitary

 Calling Sequence IsOrthogonal(A, options) IsUnitary(A, options)

Parameters

 A - square Matrix options - (optional) parameters; for a complete list, see LinearAlgebra[IsOrthogonal]

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right):$
 > $G≔\mathrm{RotationMatrix}\left(\frac{\mathrm{\pi }}{7}\right)$
 ${G}{≔}\left[\begin{array}{cc}{\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}}{{7}}\right)& {-}{\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}}{{7}}\right)\\ {\mathrm{sin}}{}\left(\frac{{\mathrm{\pi }}}{{7}}\right)& {\mathrm{cos}}{}\left(\frac{{\mathrm{\pi }}}{{7}}\right)\end{array}\right]$ (1)
 > $\mathrm{IsOrthogonal}\left(G\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{map}\left(\mathrm{simplify},G·{G}^{\mathrm{%T}}\right)$
 $\left[\begin{array}{cc}{1}& {0}\\ {0}& {1}\end{array}\right]$ (3)
 > $Q≔⟨⟨\frac{\mathrm{sqrt}\left(10\right)\cdot 3}{10},-\frac{\mathrm{sqrt}\left(10\right)}{10}⟩|⟨\frac{\mathrm{sqrt}\left(10\right)I}{10},\frac{3\mathrm{sqrt}\left(10\right)I}{10}⟩⟩$
 ${Q}{≔}\left[\begin{array}{cc}\frac{{3}{}\sqrt{{10}}}{{10}}& \frac{{I}}{{10}}{}\sqrt{{10}}\\ {-}\frac{\sqrt{{10}}}{{10}}& \frac{{3}{}{I}}{{10}}{}\sqrt{{10}}\end{array}\right]$ (4)
 > $\mathrm{IsOrthogonal}\left(Q\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{IsUnitary}\left(Q\right)$
 ${\mathrm{true}}$ (6)