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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

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):$
 > $G≔\mathrm{RotationMatrix}\left(\frac{\mathrm{π}}{7}\right)$
 ${G}{:=}\left[\begin{array}{cc}{\mathrm{cos}}{}\left(\frac{{1}}{{7}}{}{\mathrm{π}}\right)& {-}{\mathrm{sin}}{}\left(\frac{{1}}{{7}}{}{\mathrm{π}}\right)\\ {\mathrm{sin}}{}\left(\frac{{1}}{{7}}{}{\mathrm{π}}\right)& {\mathrm{cos}}{}\left(\frac{{1}}{{7}}{}{\mathrm{π}}\right)\end{array}\right]$ (1)
 > $\mathrm{IsOrthogonal}\left(G\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{map}\left(\mathrm{simplify},G\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{G}^{\mathrm{%T}}\right)$
 $\left[\begin{array}{rr}{1}& {0}\\ {0}& {1}\end{array}\right]$ (3)
 > $Q≔⟨⟨\frac{\sqrt{10}\cdot 3}{10},-\frac{\sqrt{10}}{10}⟩|⟨\frac{\sqrt{10}I}{10},\frac{3\sqrt{10}I}{10}⟩⟩$
 ${Q}{:=}\left[\begin{array}{cc}\frac{{3}}{{10}}{}\sqrt{{10}}& \frac{{1}}{{10}}{}{I}{}\sqrt{{10}}\\ {-}\frac{{1}}{{10}}{}\sqrt{{10}}& \frac{{3}}{{10}}{}{I}{}\sqrt{{10}}\end{array}\right]$ (4)
 > $\mathrm{IsOrthogonal}\left(Q\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{IsUnitary}\left(Q\right)$
 ${\mathrm{true}}$ (6)