.. _routines.linalg:

.. py:module:: dpnp.linalg

Linear algebra
==============

.. https://numpy.org/doc/stable/reference/routines.linalg.html

Matrix and vector products
--------------------------

.. autosummary::
   :toctree: generated/
   :nosignatures:

   dpnp.dot
   dpnp.linalg.multi_dot
   dpnp.vdot
   dpnp.vecdot
   dpnp.linalg.vecdot (Array API compatible)
   dpnp.inner
   dpnp.outer
   dpnp.matmul
   dpnp.linalg.matmul (Array API compatible)
   dpnp.matvec
   dpnp.vecmat
   dpnp.tensordot
   dpnp.linalg.tensordot (Array API compatible)
   dpnp.einsum
   dpnp.einsum_path
   dpnp.linalg.matrix_power
   dpnp.kron
   dpnp.linalg.cross (Array API compatible)

Decompositions
--------------

.. autosummary::
   :toctree: generated/
   :nosignatures:

   dpnp.linalg.cholesky
   dpnp.linalg.outer
   dpnp.linalg.qr
   dpnp.linalg.lu_factor
   dpnp.linalg.svd
   dpnp.linalg.svdvals

Matrix eigenvalues
------------------

.. autosummary::
   :toctree: generated/
   :nosignatures:

   dpnp.linalg.eig
   dpnp.linalg.eigh
   dpnp.linalg.eigvals
   dpnp.linalg.eigvalsh

Norms and other numbers
-----------------------

.. autosummary::
   :toctree: generated/
   :nosignatures:

   dpnp.linalg.norm
   dpnp.linalg.matrix_norm (Array API compatible)
   dpnp.linalg.vector_norm (Array API compatible)
   dpnp.linalg.cond
   dpnp.linalg.det
   dpnp.linalg.matrix_rank
   dpnp.linalg.slogdet
   dpnp.trace
   dpnp.linalg.trace (Array API compatible)

Solving linear equations
--------------------------

.. autosummary::
   :toctree: generated/
   :nosignatures:

   dpnp.linalg.solve
   dpnp.linalg.tensorsolve
   dpnp.linalg.lstsq
   dpnp.linalg.lu_solve
   dpnp.linalg.inv
   dpnp.linalg.pinv
   dpnp.linalg.tensorinv

Other matrix operations
-----------------------
.. autosummary::
   :toctree: generated/
   :nosignatures:

   dpnp.diagonal
   dpnp.linalg.diagonal (Array API compatible)
   dpnp.linalg.matrix_transpose (Array API compatible)

Exceptions
----------
.. autosummary::
   :toctree: generated/
   :nosignatures:

   dpnp.linalg.LinAlgError

Linear algebra on several matrices at once
------------------------------------------

Several of the linear algebra routines listed above are able to compute results
for several matrices at once, if they are stacked into the same array.

This is indicated in the documentation via input parameter specifications such
as ``a : (..., M, M) {dpnp.ndarray, usm_ndarray}``. This means that if for
instance given an input array ``a.shape == (N, M, M)``, it is interpreted as a
"stack" of N matrices, each of size M-by-M. Similar specification applies to
return values, for instance the determinant has ``det : (...)`` and will in
this case return an array of shape ``det(a).shape == (N,)``. This generalizes
to linear algebra operations on higher-dimensional arrays: the last 1 or 2
dimensions of a multidimensional array are interpreted as vectors or matrices,
as appropriate for each operation.