Let $A \= \left[A_1\, A_2\, ..\. \, A_m\right]$be a list of square matrices. These query commands return true if for each $i\, F\left(A_i\right) \= A_{k_i}$ for some $k_i \,$where$F\left(A\right) \= \bar{A} \($conjugation) or $F\left(A\right) \= A^t \($transposition) or $F\left(A\right) \= A^{†}\($Hermitian transposition).

With the keyword option method = "span", these commands return true if for each  $i\, F\left(A_i\right) \in$span($A$).

These commands are useful in the study of classical semi-simple Lie algebras. For example, if a semi-simple matrix Lie algebra is given, then a Cartan decomposition can easily be computed if the algebra is closed under Hermitian transposition. As another example, to calculate the Satake diagram for a non-compact simple Lie algebra, one must use a set of positive roots closed under complex conjugation.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{A1}≔\left[⟨\mathrm{I}\,-\mathrm{I}\,1⟩\,⟨0\,1\,0⟩\,⟨-\mathrm{I}\,\mathrm{I}\,1⟩\right]$

$\mathrm{Query}\left(\mathrm{A1}\,''ClosedUnderConjugation''\right)$

$\mathrm{A2}≔\mathrm{map}\left(\mathrm{Matrix}\,\left[\left[\left[1\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,-1\,0\right]\,\left[0\,0\,0\,0\right]\right]\,\left[\left[0\,1\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,-1\,0\right]\right]\,\left[\left[0\,0\,0\,0\right]\,\left[1\,0\,0\,0\right]\,\left[0\,0\,0\,-1\right]\,\left[0\,0\,0\,0\right]\right]\,\left[\left[0\,0\,0\,0\right]\,\left[0\,1\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,-1\right]\right]\,\left[\left[0\,0\,0\,-1\right]\,\left[0\,0\,1\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\right]\,\left[\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,-1\,0\,0\right]\,\left[1\,0\,0\,0\right]\right]\right]\right)$

$\mathrm{Query}\left(\mathrm{A2}\,''ClosedUnderTransposition''\right)$

$\mathrm{A3}≔\mathrm{map}\left(\mathrm{Matrix}\,\left[\left[\left[0\,-1\,0\,0\right]\,\left[1\,0\,0\,0\right]\,\left[0\,0\,0\,-1\right]\,\left[0\,0\,1\,0\right]\right]\,\left[\left[0\,-\mathrm{I}\,0\,0\right]\,\left[\mathrm{I}\,0\,0\,0\right]\,\left[0\,0\,0\,\mathrm{I}\right]\,\left[0\,0\,-\mathrm{I}\,0\right]\right]\,\left[\left[0\,0\,1\,0\right]\,\left[0\,0\,0\,0\right]\,\left[-1\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\right]\,\left[\left[0\,0\,0\,1\right]\,\left[0\,0\,1\,0\right]\,\left[0\,-1\,0\,0\right]\,\left[-1\,0\,0\,0\right]\right]\,\left[\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,1\right]\,\left[0\,0\,0\,0\right]\,\left[0\,-1\,0\,0\right]\right]\,\left[\left[0\,0\,0\,-\mathrm{I}\right]\,\left[0\,0\,\mathrm{I}\,0\right]\,\left[0\,-\mathrm{I}\,0\,0\right]\,\left[\mathrm{I}\,0\,0\,0\right]\right]\right]\right)$

$\mathrm{Query}\left(\mathrm{A3}\,''ClosedUnderHermitianTransposition''\right)$

$\mathrm{Query}\left(\mathrm{A3}\,\mathrm{method}=''span''\,''ClosedUnderHermitianTransposition''\right)$