sqfa.linalg

Utility functions for matrix algebra.

Functions

`conjugate_matrix`(A, B)	Conjugate matrix A by B, i.e. compute B A B^T.
`generalized_eigenvalues`(A, B)	Compute the generalized eigenvalues of the pair of symmetric positive definite matrices (A, B).
`generalized_eigenvectors`(A, B)	Compute the generalized eigenvectors of the pair of symmetric positive definite matrices (A, B).
`spd_inv_sqrt`(M)	For symmetric positive definite matrix M, compute the inverse square root of M.
`spd_log`(M)	Compute the matrix logarithm of a symmetric positive definite matrix.
`spd_sqrt`(M)	Compute the square root of a symmetric positive definite matrix.

sqfa.linalg.conjugate_matrix(A, B)

Conjugate matrix A by B, i.e. compute B A B^T.

Parameters:

A (torch.Tensor) – Matrix A. Shape (n_batch_A, n_dim, n_dim).
B (torch.Tensor) – Matrix B. Shape (n_batch_B, n_dim, n_out).

Returns:

C – The conjugated matrix. Shape (n_batch_A, n_batch_B, n_out, n_out). If a batch dimension is 1, it is squeezed out.

Return type:

torch.Tensor

sqfa.linalg.generalized_eigenvalues(A, B)

Compute the generalized eigenvalues of the pair of symmetric positive definite matrices (A, B).

Parameters:

A (torch.Tensor) – Symmetric positive definite matrix. Shape (n_batch_A, n_dim, n_dim).
B (torch.Tensor) – Symmetric positive definite matrix. Shape (n_batch_B, n_dim, n_dim).

Returns:

eigenvalues – The generalized eigenvalues of the pair (A, B), sorted in descending order. Shape (n_batch_A, n_batch_B, n_dim). If a batch dimension is 1, it is squeezed out.

Return type:

torch.Tensor

sqfa.linalg.generalized_eigenvectors(A, B)

Compute the generalized eigenvectors of the pair of symmetric positive definite matrices (A, B).

Parameters:

A (torch.Tensor) – Symmetric positive definite matrix. Shape (n_batch_A, n_dim, n_dim).
B (torch.Tensor) – Symmetric positive definite matrix. Shape (n_batch_B, n_dim, n_dim).

Returns:

eigenvectors (torch.Tensor) – The generalized eigenvectors sorted in descending order of the eigenvalues. Shape (n_batch_A, n_batch_B, n_dim, n_dim). If a batch dimension is 1, it is squeezed out.
eigenvalues (torch.Tensor) – The generalized eigenvalues sorted in descending order. Shape (n_batch_A, n_batch_B, n_dim). If a batch dimension is 1, it is squeezed out.

sqfa.linalg.spd_inv_sqrt(M)

For symmetric positive definite matrix M, compute the inverse square root of M.

Parameters:: M (torch.Tensor) – Symmetric positive definite matrices. Shape (n_batch, n_dim, n_dim).
Returns:: M_inv_sqrt – Inverse square root of M. Shape (n_batch, n_dim, n_dim).
Return type:: torch.Tensor

sqfa.linalg.spd_log(M)

Compute the matrix logarithm of a symmetric positive definite matrix.

Parameters:: M (torch.Tensor) – Symmetric positive definite matrices. Shape (…, n_dim, n_dim).
Returns:: M_log – The matrix logarithm of M. Shape (…, n_dim, n_dim).
Return type:: torch.Tensor

sqfa.linalg.spd_sqrt(M)

Compute the square root of a symmetric positive definite matrix.

Computes the symmetric positive definite matrix S such that SS = M.

Parameters:: M (torch.Tensor) – Symmetric positive definite matrices. Shape (…, n_dim, n_dim).
Returns:: M_sqrt – The square root of M. Shape (…, n_dim, n_dim).
Return type:: torch.Tensor