# pdParTrans: Riemannian HPD parallel transport In pdSpecEst: An Analysis Toolbox for Hermitian Positive Definite Matrices

## 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 \insertCiteC18pdSpecEst. 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

 `1` ```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

\insertAllCited

`Expm, Logm`
 ```1 2 3 4 5 6 7 8``` ```## 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 ```