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Symmetric positive definite matrices of fixed determinant

Manifolds.SPDFixedDeterminant โ€” Type
SPDFixedDeterminant{T,D} <: AbstractDecoratorManifold{โ„}

The manifold of symmetric positive definite matrices of fixed determinant $d > 0$, i.e.

\[\mathcal P_d(n) = \bigl\{ p โˆˆ โ„^{nร—n} \ \big|\ a^\mathrm{T}pa > 0 \text{ for all } a โˆˆ โ„^{n}\backslash\{0\} \text{ and } \det(p) = d \bigr\}.\]

This manifold is modelled as a submanifold of SymmetricPositiveDefinite(n), see EmbeddedSubmanifoldType for the implications, but for example retractions and inverse retractions are all available

These matrices are sometimes also called isochoric, which refers to the interpretation of the matrix representing an ellipsoid. All ellipsoids that represent points on this manifold have the same volume.

The tangent space is modelled the same as for SymmetricPositiveDefinite(n) and consists of all symmetric matrices $X$ whose product with $p^{-1}$ has zero trace

\[ T_p\mathcal P_d(n) = \bigl\{ X \in \mathbb R^{nร—n} \big|\ X=X^\mathrm{T} \text{ and } \operatorname{tr}(p^{-1}X) = 0 \bigr\},\]

since for a constant determinant we require that $0 = D\det(p)[X] = \det(p)\operatorname{tr}(p^{-1}X)$ for all tangent vectors $X$. At the identity matrix this is the set of symmetric matrices of zero trace.

Constructor

SPDFixedDeterminant(n::Int, d::Real=1.0; parameter::Symbol=:type)

Generate the manifold $\mathcal P_d(n) \subset \mathcal P(n)$ of determinant $d$, which defaults to 1.0.

parameter: whether a type parameter should be used to store n. By default size is stored in type. Value can either be :field or :type.

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This manifold is a submanifold of the symmetric positive definite matrices and hence inherits most properties therefrom.

The differences are the functions

ManifoldsBase.check_vector โ€” Method
check_vector(M::SPDFixedDeterminant, p, X; atol, rtol, kwargs... )

Check whether X is a tangent vector to manifold point p on the SPDFixedDeterminant M. X has to be a tangent vector on SymmetricPositiveDefinite, so a symmetric matrix, and additionally fulfill $\operatorname{tr}(p^{-1}X) = 0$ up to max(atol, rtol * sqrt(n) * norm(M, p, X)). The relative tolerance rtol refers to the size of X; its default is the one of isapprox.

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