pyriemann.utils.base.ddexpm

pyriemann.utils.base.ddexpm(X, Cref)

Directional derivative of the matrix exponential.

The directional derivative of the matrix exponential at a SPD/HPD matrix \(\mathbf{C}_{\text{ref}}\) in the direction of a SPD/HPD matrix \(\mathbf{X}\) is defined as Eq. (V.13) in [1]:

\[\text{ddexpm}(\mathbf{X}, \mathbf{C}_{\text{ref}}) = \mathbf{V} \left( \text{fddexpm}(\mathbf{\Lambda}) \odot \mathbf{V}^H \mathbf{X} \mathbf{V} \right) \mathbf{V}^H\]

where \(\mathbf{\Lambda}\) is the diagonal matrix of eigenvalues of and \(\mathbf{V}\) the eigenvectors of \(\mathbf{C}_{\text{ref}}\), and \(\text{fddexpm}\) the first divided difference of the exponential function.

Parameters:

Xndarray, shape (…, n, n)

SPD/HPD matrices.

Crefndarray, shape (n, n)

SPD/HPD matrix.

Returns:

ddexpmndarray, shape (…, n, n)

Directional derivative of the matrix exponential.

Notes

Added in version 0.8.

References

[1]

Matrix Analysis R. Bhatia, Springer, 1997