pdParTrans {pdSpecEst}

Riemannian HPD parallel transport

Description

pdParTrans computes the parallel transport on the manifold of HPD matrices equipped with the affine-invariant Riemannian metric as described in e.g., Chapter 2 of (Chau 2018). That is, the function computes the parallel transport of a Hermitian matrix W in the tangent space at the HPD matrix P along a geodesic curve in the direction of the Hermitian matrix V in the tangent space at P for a unit time step.

Usage

pdParTrans(P, V, W)

Arguments

P

a (d,d)-dimensional HPD matrix.

V

a (d,d)-dimensional Hermitian matrix corresponding to a vector in the tangent space of P.

W

a (d,d)-dimensional Hermitian matrix corresponding to a vector in the tangent space of P.

Value

a (d,d)-dimensional Hermitian matrix corresponding to the parallel transportation of W in the direction of V along a geodesic curve for a unit time step.

References

Chau J (2018). Advances in Spectral Analysis for Multivariate, Nonstationary and Replicated Time Series. phdthesis, Universite catholique de Louvain.

See Also

Expm, Logm

Examples

## Transport the vector W to the tangent space at the identity
W <- matrix(complex(real = rnorm(9), imaginary = rnorm(9)), nrow = 3)
diag(W) <- rnorm(3)
W[lower.tri(W)] <- t(Conj(W))[lower.tri(W)]
p <- matrix(complex(real = rnorm(9), imaginary = rnorm(9)), nrow = 3)
P <- t(Conj(p)) %*% p

pdParTrans(P, Logm(P, diag(3)), W) ## whitening transport