inversion_ideas.operators.get_diagonal

inversion_ideas.operators.get_diagonal(operator)

Extract diagonal of a linear operator.

Extracts the main diagonal of a square linear operator. If the operator is a dense or sparse array with a diagonal method, the method will be used. For LinearOperator`s or any other operator that doesn't implement the ``diagonal` method, the diagonal will be computed by N projections using the unit vectors of the standard basis of \(\mathcal{R}^N\).

Important

Extracting the diagonal of a LinearOperator requires computing multiple dot products. This can be quite expensive for large operators.

Parameters:

operator(n, n) array, sparse array or LinearOperator

Square linear operator from which the diagonal will be extracted.

Returns:

diagonal(n,) array

1D array containing the diagonal of the linear operator.

Notes

Consider a \(N \times N\) linear operator \(A\), and let \(\{ e_i \}_1^N\) be the standard basis for \(\mathcal{R}^N\). The \(i\)-th diagonal element of \(A\) (\(a_{ii}\)) can be obtained as:

\[a_{ii} = e_i^T A e_i\]

Examples

>>> import numpy as np
>>> from scipy.sparse.linalg import aslinearoperator

Build a linear operator from a matrix, and extract its diagonal:

>>> a_matrix = np.array(
...     [
...      [1., 2., 3.],
...      [4., 5., 6.],
...      [7., 8., 9.],
...     ]
... )
>>> linop = aslinearoperator(a_matrix)
>>> get_diagonal(linop)
array([1., 5., 9.])