The general linear group

GeneralLinearGroup{𝔽,T}

The general linear group $\mathrm{GL}(n)$ is the set of all invertible matrices

\[\mathrm{GL}(n) = \bigl\{ g ∈ 𝔽^{n×n}\ \big|\ \mathrm{det}(p) ≠ 0\bigr \}, \qquad 𝔽 ∈ \{ ℝ, ℂ \},\]

equipped with the MatrixMultiplicationGroupOperation as the group operation.

The set of invertible matrices is a Riemannian manifold, since it inherits its structure from the embedding as an open subset of the space of matrices $ℝ^{n×n}$.

Constructor

GeneralLinearGroup(n::Int; field=ℝ, kwargs...)

Generate the general linear group group on $𝔽^{n×n}$. All keyword arguments in kwargs... are passed on to InvertibleMatrices.

exp(::GeneralLinearGroup, ::Identity{MatrixMultiplicationGroupOperation}, X)
exp!(::GeneralLinearGroup, g, ::Identity{MatrixMultiplicationGroupOperation}, X)

Compute the Lie group exponential on the GeneralLinearGroup, which is given by the matrix exponential

\[\exp X = \sum_{k=0}^{∞} \frac{1}{k!}X^k\]

see also [HN12, Example 9.2.3 (b)]

exp(::GeneralLinearGroup, ::Identity{MatrixMultiplicationGroupOperation}, X)
exp!(::GeneralLinearGroup, g, ::Identity{MatrixMultiplicationGroupOperation}, X)

Compute the Lie group exponential on the GeneralLinearGroup, which is given by the matrix exponential

\[\exp X = \sum_{k=0}^{∞} \frac{1}{k!}X^k\]

see also [HN12, Example 9.2.3 (b)]