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# [The Symmetric Positive Definite $n\times n$ Matrices](@id SymmetricPositiveDefiniteManifold)

The manifold of symmetric positive definite matrices $\mathcal P(3) = \bigl{ A \in \mathbb R^{n\times n}\ \big|\ A = A^{\mathrm{T}} \text{ and } x^{\mathrm{T}}Ax > 0 \text{ for } 0\neq x \in\mathbb R^n \bigr}$ posesses the following instances of the abstract types Manifold, MPoint, and TVector.

Modules = [Manopt]
Pages = ["manifolds/SymmetricPositiveDefinite.jl"]
Order = [:type]

While there are two Riemannian metrics available, this one focuses on the affine metric. The Log-Euclidean Metric needs at least a new tangent vector type inheriting from <: TVector.

Note that saving the points on the manifold in this format is a little bit rendundant, it' enough to save the upper triangular matrix. For ease of computations this is – for now – adapted from Matlab.

# Functions

Modules = [Manopt]
Pages = ["manifolds/SymmetricPositiveDefinite.jl"]
Order = [:function]
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