tensor.nlinalg
– Linear Algebra Ops Using Numpy¶
Note
This module is not imported by default. You need to import it to use it.
API¶

aesara.tensor.nlinalg.
matrix_dot
(*args)[source]¶ Shorthand for product between several dots.
Given matrices ,
matrix_dot
will generate the matrix product between all in the given order, namely .

aesara.tensor.nlinalg.
matrix_power
(M, n)[source]¶ Raise a square matrix,
M
, to the (integer) powern
.This implementation uses exponentiation by squaring which is significantly faster than the naive implementation. The time complexity for exponentiation by squaring is :math:
mathcal{O}((n log M)^k)
Parameters:  M (TensorVariable) –
 n (int) –

aesara.tensor.nlinalg.
pinv
(x, hermitian=False)[source]¶ Computes the pseudoinverse of a matrix .
The pseudoinverse of a matrix , denoted , is defined as: “the matrix that ‘solves’ [the leastsquares problem] ,” i.e., if is said solution, then is that matrix such that .
Note that , so is close to the identity matrix. This method is not faster than
matrix_inverse
. Its strength comes from that it works for nonsquare matrices. If you have a square matrix though,matrix_inverse
can be both more exact and faster to compute. Also this op does not get optimized into a solve op.

aesara.tensor.nlinalg.
qr
(a, mode='reduced')[source]¶ Computes the QR decomposition of a matrix. Factor the matrix a as qr, where q is orthonormal and r is uppertriangular.
Parameters:  a (array_like, shape (M, N)) – Matrix to be factored.
 mode ({'reduced', 'complete', 'r', 'raw'}, optional) –
If K = min(M, N), then
 ’reduced’
 returns q, r with dimensions (M, K), (K, N)
 ’complete’
 returns q, r with dimensions (M, M), (M, N)
 ’r’
 returns r only with dimensions (K, N)
 ’raw’
 returns h, tau with dimensions (N, M), (K,)
Note that array h returned in ‘raw’ mode is transposed for calling Fortran.
Default mode is ‘reduced’
Returns:  q (matrix of float or complex, optional) – A matrix with orthonormal columns. When mode = ‘complete’ the result is an orthogonal/unitary matrix depending on whether or not a is real/complex. The determinant may be either +/ 1 in that case.
 r (matrix of float or complex, optional) – The uppertriangular matrix.

aesara.tensor.nlinalg.
svd
(a, full_matrices=1, compute_uv=1)[source]¶ This function performs the SVD on CPU.
Parameters:  full_matrices (bool, optional) – If True (default), u and v have the shapes (M, M) and (N, N), respectively. Otherwise, the shapes are (M, K) and (K, N), respectively, where K = min(M, N).
 compute_uv (bool, optional) – Whether or not to compute u and v in addition to s. True by default.
Returns: U, V, D
Return type: matrices

aesara.tensor.nlinalg.
tensorinv
(a, ind=2)[source]¶ Does not run on GPU; Aesara utilization of numpy.linalg.tensorinv;
Compute the ‘inverse’ of an Ndimensional array. The result is an inverse for
a
relative to the tensordot operationtensordot(a, b, ind)
, i. e., up to floatingpoint accuracy,tensordot(tensorinv(a), a, ind)
is the “identity” tensor for the tensordot operation.Parameters:  a (array_like) – Tensor to ‘invert’. Its shape must be ‘square’, i. e.,
prod(a.shape[:ind]) == prod(a.shape[ind:])
.  ind (int, optional) – Number of first indices that are involved in the inverse sum. Must be a positive integer, default is 2.
Returns: b –
a
’s tensordot inverse, shapea.shape[ind:] + a.shape[:ind]
.Return type: ndarray
Raises: LinAlgError – If
a
is singular or not ‘square’ (in the above sense). a (array_like) – Tensor to ‘invert’. Its shape must be ‘square’, i. e.,

aesara.tensor.nlinalg.
tensorsolve
(a, b, axes=None)[source]¶ Aesara utilization of numpy.linalg.tensorsolve. Does not run on GPU!
Solve the tensor equation
a x = b
for x. It is assumed that all indices ofx
are summed over in the product, together with the rightmost indices ofa
, as is done in, for example,tensordot(a, x, axes=len(b.shape))
.Parameters:  a (array_like) – Coefficient tensor, of shape
b.shape + Q
.Q
, a tuple, equals the shape of that subtensor ofa
consisting of the appropriate number of its rightmost indices, and must be such thatprod(Q) == prod(b.shape)
(in which sensea
is said to be ‘square’).  b (array_like) – Righthand tensor, which can be of any shape.
 axes (tuple of ints, optional) – Axes in
a
to reorder to the right, before inversion. If None (default), no reordering is done.
Returns: x
Return type: ndarray, shape Q
Raises: LinAlgError – If
a
is singular or not ‘square’ (in the above sense). a (array_like) – Coefficient tensor, of shape