class scipy.sparse.linalg.LinearOperator(dtype, shape)[source]

Common interface for performing matrix vector products

Many iterative methods (e.g. cg, gmres) do not need to know the individual entries of a matrix to solve a linear system A*x=b. Such solvers only require the computation of matrix vector products, A*v where v is a dense vector. This class serves as an abstract interface between iterative solvers and matrix-like objects.

To construct a concrete LinearOperator, either pass appropriate callables to the constructor of this class, or subclass it.

A subclass must implement either one of the methods _matvec and _matmat, and the attributes/properties shape (pair of integers) and dtype (may be None). It may call the __init__ on this class to have these attributes validated. Implementing _matvec automatically implements _matmat (using a naive algorithm) and vice-versa.

Optionally, a subclass may implement _rmatvec or _adjoint to implement the Hermitian adjoint (conjugate transpose). As with _matvec and _matmat, implementing either _rmatvec or _adjoint implements the other automatically. Implementing _adjoint is preferable; _rmatvec is mostly there for backwards compatibility.


shape : tuple

Matrix dimensions (M,N).

matvec : callable f(v)

Returns returns A * v.

rmatvec : callable f(v)

Returns A^H * v, where A^H is the conjugate transpose of A.

matmat : callable f(V)

Returns A * V, where V is a dense matrix with dimensions (N,K).

dtype : dtype

Data type of the matrix.

See also

Construct LinearOperators


The user-defined matvec() function must properly handle the case where v has shape (N,) as well as the (N,1) case. The shape of the return type is handled internally by LinearOperator.

LinearOperator instances can also be multiplied, added with each other and exponentiated, all lazily: the result of these operations is always a new, composite LinearOperator, that defers linear operations to the original operators and combines the results.


>>> import numpy as np
>>> from scipy.sparse.linalg import LinearOperator
>>> def mv(v):
...     return np.array([2*v[0], 3*v[1]])
>>> A = LinearOperator((2,2), matvec=mv)
>>> A
<2x2 LinearOperator with unspecified dtype>
>>> A.matvec(np.ones(2))
array([ 2.,  3.])
>>> A * np.ones(2)
array([ 2.,  3.])


args (tuple) For linear operators describing products etc. of other linear operators, the operands of the binary operation.


adjoint() Hermitian adjoint.
dot(x) Matrix-matrix or matrix-vector multiplication.
matmat(X) Matrix-matrix multiplication.
matvec(x) Matrix-vector multiplication.
rmatvec(x) Adjoint matrix-vector multiplication.
transpose() Transpose this linear operator.