cartan
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Symmetric Positive Definite Matrices SPD(N)

The open cone of $N \times N$ symmetric positive definite matrices; a Cartan–Hadamard manifold (nonpositive curvature) with a rich geometric structure.

ManifoldRetractionConnectionParallelTransportVectorTransportGeodesicInterp.

Coming Soon

SPD(N) with affine-invariant metric is planned for a future release.

Geometry Preview

PropertyValue
PointsSymmetric positive definite matrices
Tangent space at $P$ (symmetric)
Metric (affine-invariant)
Dimension$N(N+1)/2$
Sectional curvature$K \leq 0$ (Cartan–Hadamard)
Cut locusEmpty: globally unique geodesics

Exponential map (affine-invariant):

Key property: Nonpositive curvature means the Fréchet mean is unique and gradient descent converges globally; no cut locus complications.

Applications

  • Diffusion tensor imaging: SPD(3) diffusion tensors
  • Covariance estimation: sample covariance matrices as points on SPD
  • Brain-computer interfaces: Riemannian classification of EEG covariances
  • Finance: correlation matrices in portfolio optimization