numpy.linalg.inv — NumPy v2.4 Manual (original) (raw)

linalg.inv(a)[source]#

Compute the inverse of a matrix.

Given a square matrix a, return the matrix ainv satisfyinga @ ainv = ainv @ a = eye(a.shape[0]).

Parameters:

a(…, M, M) array_like

Matrix to be inverted.

Returns:

ainv(…, M, M) ndarray or matrix

Inverse of the matrix a.

Raises:

LinAlgError

If a is not square or inversion fails.

Notes

Broadcasting rules apply, see the numpy.linalg documentation for details.

If a is detected to be singular, a LinAlgError is raised. If a is ill-conditioned, a LinAlgError may or may not be raised, and results may be inaccurate due to floating-point errors.

References

Examples

import numpy as np from numpy.linalg import inv a = np.array([[1., 2.], [3., 4.]]) ainv = inv(a) np.allclose(a @ ainv, np.eye(2)) True np.allclose(ainv @ a, np.eye(2)) True

If a is a matrix object, then the return value is a matrix as well:

ainv = inv(np.matrix(a)) ainv matrix([[-2. , 1. ], [ 1.5, -0.5]])

Inverses of several matrices can be computed at once:

a = np.array([[[1., 2.], [3., 4.]], [[1, 3], [3, 5]]]) inv(a) array([[[-2. , 1. ], [ 1.5 , -0.5 ]], [[-1.25, 0.75], [ 0.75, -0.25]]])

If a matrix is close to singular, the computed inverse may not satisfya @ ainv = ainv @ a = eye(a.shape[0]) even if a LinAlgErroris not raised:

a = np.array([[2,4,6],[2,0,2],[6,8,14]]) inv(a) # No errors raised array([[-1.12589991e+15, -5.62949953e+14, 5.62949953e+14], [-1.12589991e+15, -5.62949953e+14, 5.62949953e+14], [ 1.12589991e+15, 5.62949953e+14, -5.62949953e+14]]) a @ inv(a) array([[ 0. , -0.5 , 0. ], # may vary [-0.5 , 0.625, 0.25 ], [ 0. , 0. , 1. ]])

To detect ill-conditioned matrices, you can use numpy.linalg.cond to compute its condition number [1]. The larger the condition number, the more ill-conditioned the matrix is. As a rule of thumb, if the condition number cond(a) = 10**k, then you may lose up to k digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods.

from numpy.linalg import cond cond(a) np.float64(8.659885634118668e+17) # may vary

It is also possible to detect ill-conditioning by inspecting the matrix’s singular values directly. The ratio between the largest and the smallest singular value is the condition number:

from numpy.linalg import svd sigma = svd(a, compute_uv=False) # Do not compute singular vectors sigma.max()/sigma.min() 8.659885634118668e+17 # may vary