Query[ClosedUnderConjugation] - check if a list of vectors or matrices is closed under complex conjugation
Query[ClosedUnderTransposition] - check if a list of square matrices is closed under transposition
Query[ClosedUnderHermitianTransposition] - check if a list of square matrices is closed under Hermitian (complex conjugation) transposition
Query(A, options , ClosedUnderConjugation)
Query(A, options , ClosedUnderTransposition)
Query(A, options , ClosedUnderHermitianTransposition)
A - a list of column vectors, or a list of square matrices
option - the keyword method
Let A = A1, A2, ... , Am be a list of square matrices. These query commands return true if for each i, FAi = Aki for some ki , where FA = A‾ (conjugation) or FA = At (transposition) or FA = A†(Hermitian transposition).
With the keyword option method = "span", these commands return true if for each i, FAi ∈ 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.
We check that a list of vectors is closed under complex-conjugation.
A1 ≔ I,−I,1,0,1,0,−I,I,1
We check that the list of matrices defining so2,2 is closed under transposition.
A2 ≔ map⁡Matrix,1,0,0,0,0,0,0,0,0,0,−1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,−1,0,0,0,0,0,1,0,0,0,0,0,0,−1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,−1,0,0,0,−1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,−1,0,0,1,0,0,0
We check that the span of the matrices defining so∗4 is closed under Hermitian transposition.
A3 ≔ map⁡Matrix,0,−1,0,0,1,0,0,0,0,0,0,−1,0,0,1,0,0,−I,0,0,I,0,0,0,0,0,0,I,0,0,−I,0,0,0,1,0,0,0,0,0,−1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,−1,0,0,−1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,−1,0,0,0,0,0,−I,0,0,I,0,0,−I,0,0,I,0,0,0
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