Quaternionic unitary group

[mateuszbaran]

This should solve #382 . exp and log are currently only supported for the n=1 case because I couldn't find a generic quaternionic matrix exponential and logarithm.

Also, plain numbers should also work for QuaternionicUnitary(1).

[sethaxen]

exp and log are currently only supported for the n=1 case because I couldn't find a generic quaternionic matrix exponential and logarithm.

They can be supported by expanding each quaternion into its 2x2 complex matrix representation, which become blocks in a matrix with nxn blocks, so an nxn quaternionic matrix corresponds to an 2nx2n complex matrix. This transform preserves the matrix product, so the result is a closed subgroup of U(2n), on which one can compute exp/log and then map back to quaternions.

But I think it's best to handle this kind of thing at the level of the quaternion implementation, not here. e.g. Quaternions.jl had JuliaGeometry/Quaternions.jl#46 for adding support for matrix exponential, and this could be expanded to support all matrix functions. Are there fallbacks to for exp_lie and log_lie that call the matrix exponential? If so, then this would already be supported with such a PR to Quaternions.

Why QuaternionicUnitary(n) and not something like Unitary(n, 𝔽=ℂ)? That would be more consistent with how we allow constructors to take fields elsewhere.

[codecov]

Codecov Report

Merging #506 (fa488a5) into master (0cd7158) will increase coverage by 0.00%.
The diff coverage is 100.00%.

@@           Coverage Diff           @@
##           master     #506   +/-   ##
=======================================
  Coverage   98.87%   98.87%           
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  Files          96       96           
  Lines        8714     8735   +21     
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+ Hits         8616     8637   +21     
  Misses         98       98           

Impacted Files	Coverage Δ
src/Manifolds.jl	100.00% <ø> (ø)
src/groups/unitary.jl	100.00% <100.00%> (ø)
src/manifolds/GeneralUnitaryMatrices.jl	100.00% <100.00%> (ø)
src/manifolds/Unitary.jl	100.00% <100.00%> (ø)

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[mateuszbaran]

They can be supported by expanding each quaternion into its 2x2 complex matrix representation, which become blocks in a matrix with nxn blocks, so an nxn quaternionic matrix corresponds to an 2nx2n complex matrix. This transform preserves the matrix product, so the result is a closed subgroup of U(2n), on which one can compute exp/log and then map back to quaternions.

Thanks, I didn't know that.

But I think it's best to handle this kind of thing at the level of the quaternion implementation, not here. e.g. Quaternions.jl had JuliaGeometry/Quaternions.jl#46 for adding support for matrix exponential, and this could be expanded to support all matrix functions. Are there fallbacks to for exp_lie and log_lie that call the matrix exponential? If so, then this would already be supported with such a PR to Quaternions.

OK, so I won't work on that here in this PR. Currently there are no such fallbacks but they will be easy to add when quaternionic matrix exp is available.

Why QuaternionicUnitary(n) and not something like Unitary(n, 𝔽=ℂ)? That would be more consistent with how we allow constructors to take fields elsewhere.

Mostly to be consistent with the existing Orthogonal/Unitary. I don't have a strong opinion about it but maybe let's ask @kellertuer what he prefers.

[kellertuer]

Why QuaternionicUnitary(n) and not something like Unitary(n, 𝔽=ℂ)? That would be more consistent with how we allow constructors to take fields elsewhere.

Mostly to be consistent with the existing Orthogonal/Unitary. I don't have a strong opinion about it but maybe let's ask @kellertuer what he prefers.

I think for Orthogonal and Unitary there is different terms and names, so that's why I preferred to split that, but for complex or quaternionic Unitary the field should be fine.