# Orthogonal and Unitary matrices

Both OrthogonalMatrices and UnitaryMatrices are quite similar, as are Rotations, as well as unitary matrices with determinant equal to one. So these share a {common implementation}(@ref generalunitarymatrices)

## Orthogonal Matrices

Manifolds.OrthogonalMatricesType
 OrthogonalMatrices{n} = GeneralUnitaryMatrices{n,ℝ,AbsoluteDeterminantOneMatrices}

The manifold of (real) orthogonal matrices $\mathrm{O}(n)$.

OrthogonalMatrices(n)
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## Unitary Matrices

Manifolds.UnitaryMatricesType
const UnitaryMatrices{n,𝔽} = AbstarctUnitaryMatrices{n,𝔽,AbsoluteDeterminantOneMatrices}

The manifold $U(n,𝔽)$ of $n×n$ complex matrices (when 𝔽=ℂ) or quaternionic matrices (when 𝔽=ℍ) such that

$p^{\mathrm{H}}p = \mathrm{I}_n,$

where $\mathrm{I}_n$ is the $n×n$ identity matrix. Such matrices p have a property that $\lVert \det(p) \rVert = 1$.

The tangent spaces are given by

$$$T_pU(n) \coloneqq \bigl\{ X \big| pY \text{ where } Y \text{ is skew symmetric, i. e. } Y = -Y^{\mathrm{H}} \bigr\}$$$

But note that tangent vectors are represented in the Lie algebra, i.e. just using $Y$ in the representation above.

Constructor

UnitaryMatrices(n, 𝔽::AbstractNumbers=ℂ)

see also OrthogonalMatrices for the real valued case.

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ManifoldsBase.manifold_dimensionMethod
manifold_dimension(M::UnitaryMatrices{n,ℂ}) where {n}

Return the dimension of the manifold unitary matrices.

$$$\dim_{\mathrm{U}(n)} = n^2.$$$
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ManifoldsBase.manifold_dimensionMethod
manifold_dimension(M::UnitaryMatrices{n,ℍ})

Return the dimension of the manifold unitary matrices.

$$$\dim_{\mathrm{U}(n, ℍ)} = n(2n+1).$$$
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## Common functions

Manifolds.GeneralUnitaryMatricesType
GeneralUnitaryMatrices{n,𝔽,S<:AbstractMatrixType} <: AbstractDecoratorManifold

A common parametric type for matrices with a unitary property of size $n×n$ over the field $\mathbb F$ which additionally have the AbstractMatrixType, e.g. are DeterminantOneMatrices.

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Base.expMethod
exp(M::Rotations, p, X)
exp(M::OrthogonalMatrices, p, X)
exp(M::UnitaryMatrices, p, X)

Compute the exponential map, that is, since $X$ is represented in the Lie algebra,

exp_p(X) = p\mathrm{e}^X

For different sizes, like $n=2,3,4$ there is specialised implementations

The algorithm used is a more numerically stable form of those proposed in [Gallier2002] and [Andrica2013].

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Base.logMethod
log(M::Rotations, p, X)
log(M::OrthogonalMatrices, p, X)
log(M::UnitaryMatrices, p, X)

Compute the logarithmic map, that is, since the resulting $X$ is represented in the Lie algebra,

log_p q = \log(p^{\mathrm{H}q)

which is projected onto the skew symmetric matrices for numerical stability.

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Base.logMethod
log(M::Rotations, p, q)

Compute the logarithmic map on the Rotations manifold M which is given by

$$$\log_p q = \operatorname{log}(p^{\mathrm{T}}q)$$$

where $\operatorname{Log}$ denotes the matrix logarithm. For numerical stability, the result is projected onto the set of skew symmetric matrices.

For antipodal rotations the function returns deterministically one of the tangent vectors that point at q.

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Manifolds.cos_angles_4d_rotation_matrixMethod
cos_angles_4d_rotation_matrix(R)

4D rotations can be described by two orthogonal planes that are unchanged by the action of the rotation (vectors within a plane rotate only within the plane). The cosines of the two angles $α,β$ of rotation about these planes may be obtained from the distinct real parts of the eigenvalues of the rotation matrix. This function computes these more efficiently by solving the system

\begin{aligned} \cos α + \cos β &= \frac{1}{2} \operatorname{tr}(R)\\ \cos α \cos β &= \frac{1}{8} \operatorname{tr}(R)^2 - \frac{1}{16} \operatorname{tr}((R - R^T)^2) - 1. \end{aligned}

By convention, the returned values are sorted in decreasing order. See also angles_4d_skew_sym_matrix.

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ManifoldsBase.check_pointMethod
check_point(M::Rotations, p; kwargs...)

Check whether p is a valid point on the UnitaryMatrices M, i.e. that $p$ has an determinante of absolute value one, i.e. that $p^{\mathrm{H}}p$

The tolerance for the last test can be set using the kwargs....

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ManifoldsBase.check_pointMethod
check_point(M::UnitaryMatrices, p; kwargs...)
check_point(M::OrthogonalMatrices, p; kwargs...)
check_point(M::GeneralUnitaryMatrices{n,𝔽}, p; kwargs...)

Check whether p is a valid point on the UnitaryMatrices or [OrthogonalMatrices] M, i.e. that $p$ has an determinante of absolute value one

The tolerance for the last test can be set using the kwargs....

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ManifoldsBase.check_vectorMethod
check_vector(M::UnitaryMatrices{n}, p, X; kwargs... )
check_vector(M::OrthogonalMatrices{n}, p, X; kwargs... )
check_vector(M::Rotations{n}, p, X; kwargs... )
check_vector(M::GeneralUnitaryMatrices{n,𝔽}, p, X; kwargs... )

Check whether X is a tangent vector to p on the UnitaryMatrices space M, i.e. after check_point(M,p), X has to be skew symmetric (Hermitian) and orthogonal to p.

The tolerance for the last test can be set using the kwargs....

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ManifoldsBase.embedMethod
embed(M::GeneralUnitaryMatrices{n,𝔽}, p, X)

Embed the tangent vector X at point p in M from its Lie algebra representation (set of skew matrices) into the Riemannian submanifold representation

$$$X_{\text{embedded}} = p * X$$$
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ManifoldsBase.get_coordinatesMethod
get_coordinates(M::Rotations, p, X)
get_coordinates(M::OrthogonalMatrices, p, X)
get_coordinates(M::UnitaryMatrices, p, X)

Extract the unique tangent vector components $X^i$ at point p on Rotations $\mathrm{SO}(n)$ from the matrix representation X of the tangent vector.

The basis on the Lie algebra $𝔰𝔬(n)$ is chosen such that for $\mathrm{SO}(2)$, $X^1 = θ = X_{21}$ is the angle of rotation, and for $\mathrm{SO}(3)$, $(X^1, X^2, X^3) = (X_{32}, X_{13}, X_{21}) = θ u$ is the angular velocity and axis-angle representation, where $u$ is the unit vector along the axis of rotation.

For $\mathrm{SO}(n)$ where $n ≥ 4$, the additional elements of $X^i$ are $X^{j (j - 3)/2 + k + 1} = X_{jk}$, for $j ∈ [4,n], k ∈ [1,j)$.

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ManifoldsBase.get_embeddingMethod
get_embedding(M::OrthogonalMatrices{n})
get_embedding(M::Rotations{n})
get_embedding(M::UnitaryMatrices{n})

Return the embedding, i.e. The $\mathbb F^{n×n}$, where $\mathbb F = \mathbb R$ for the first two and $\mathbb F = \mathbb C$ for the unitary matrices.

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ManifoldsBase.injectivity_radiusMethod
injectivity_radius(G::GeneraliUnitaryMatrices)

Return the injectivity radius for general unitary matrix manifolds, which is[1]

$$$\operatorname{inj}_{\mathrm{U}(n)} = π.$$$
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ManifoldsBase.injectivity_radiusMethod
injectivity_radius(G::GeneralUnitaryMatrices{n,ℂ,DeterminantOneMatrices})

Return the injectivity radius for general complex unitary matrix manifolds, where the determinant is $+1$, which is[1]

$$$\operatorname{inj}_{\mathrm{SU}(n)} = π \sqrt{2}.$$$
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ManifoldsBase.manifold_dimensionMethod
manifold_dimension(M::GeneralUnitaryMatrices{n,ℂ,DeterminantOneMatrices})

Return the dimension of the manifold of special unitary matrices.

$$$\dim_{\mathrm{SU}(n)} = n^2-1.$$$
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ManifoldsBase.manifold_dimensionMethod
manifold_dimension(M::Rotations)
manifold_dimension(M::OrthogonalMatrices)

Return the dimension of the manifold orthogonal matrices and of the manifold of rotations

$$$\dim_{\mathrm{O}(n)} = \dim_{\mathrm{SO}(n)} = \frac{n(n-1)}{2}.$$$
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ManifoldsBase.projectMethod
 project(M::OrthogonalMatrices{n}, p, X)
project(M::Rotations{n}, p, X)
project(M::UnitaryMatrices{n}, p, X)

Orthogonally project the tangent vector $X ∈ 𝔽^{n × n}$, $\mathbb F ∈ \{\mathbb R, \mathbb C\}$ to the tangent space of M at p, and change the representer to use the corresponding Lie algebra, i.e. we compute

$$$\operatorname{proj}_p(X) = \frac{p^{\mathrm{H}} X - (p^{\mathrm{H}} X)^{\mathrm{H}}}{2},$$$
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ManifoldsBase.projectMethod
 project(G::UnitaryMatrices{n}, p)
project(G::OrthogonalMatrices{n}, p)

Project the point $p ∈ 𝔽^{n × n}$ to the nearest point in $\mathrm{U}(n,𝔽)=$Unitary(n,𝔽) under the Frobenius norm. If $p = U S V^\mathrm{H}$ is the singular value decomposition of $p$, then the projection is

$$$\operatorname{proj}_{\mathrm{U}(n,𝔽)} \colon p ↦ U V^\mathrm{H}.$$$
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ManifoldsBase.retractMethod
retract(M::Rotations, p, X, ::PolarRetraction)
retract(M::OrthogonalMatrices, p, X, ::PolarRetraction)

Compute the SVD-based retraction on the Rotations and OrthogonalMatrices M from p in direction X (as an element of the Lie group) and is a second-order approximation of the exponential map. Let

$$$USV = p + pX$$$

be the singular value decomposition, then the formula reads

$$$\operatorname{retr}_p X = UV^\mathrm{T}.$$$
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ManifoldsBase.retractMethod
retract(M::Rotations, p, X, ::QRRetraction)
retract(M::OrthogonalMatrices, p. X, ::QRRetraction)

Compute the QR-based retraction on the Rotations and OrthogonalMatrices M from p in direction X (as an element of the Lie group), which is a first-order approximation of the exponential map.

This is also the default retraction on these manifolds.

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## Footnotes and References

• Gallier2002

Gallier J.; Xu D.; Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. International Journal of Robotics and Automation (2002), 17(4), pp. 1-11. pdf.

• Andrica2013

Andrica D.; Rohan R.-A.; Computing the Rodrigues coefficients of the exponential map of the Lie groups of matrices. Balkan Journal of Geometry and Its Applications (2013), 18(2), pp. 1-2. pdf.

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