scipy.linalg.

sqrtm#

scipy.linalg.sqrtm(A)[source]#

Compute, if exists, the matrix square root.

The matrix square root of A is a matrix X such that X @ X = A. Every square matrix is not guaranteed to have a matrix square root, for example, the array [[0, 1], [0, 0]] does not have a square root.

Moreover, not every real matrix has a real square root. Hence, for real-valued matrices the return type can be complex if, numerically, there is an eigenvalue on the negative real axis.

Array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

Parameters:

Andarray: Input with last two dimensions are square (..., n, n).

Returns:

sqrtmndarray: Computed matrix squareroot of A with same size (..., n, n).

Notes

This function uses the Schur decomposition method to compute the matrix square root following [1] and for real matrices [2]. Moreover, note that, there exist matrices that have square roots that are not polynomials in A. For a classical example from [2], the matrix satisfies:

[ a, a**2 + 1]**2     [-1,  0]
[-1,       -a]     =  [ 0, -1]

for any scalar a but it is not a polynomial in -I. Thus, they will not be found by this function.

References

[1]

Edvin Deadman, Nicholas J. Higham, Rui Ralha (2013) “Blocked Schur Algorithms for Computing the Matrix Square Root, Lecture Notes in Computer Science, 7782. pp. 171-182. DOI:10.1016/0024-3795(87)90118-2

[2] (1,2)

Nicholas J. Higham (1987) “Computing real square roots of a real matrix”, Linear Algebra and its Applications, 88/89:405-430. DOI:10.1016/0024-3795(87)90118-2

Examples

>>> import numpy as np
>>> from scipy.linalg import sqrtm
>>> a = np.array([[1.0, 3.0], [1.0, 4.0]])
>>> r = sqrtm(a)
>>> r
array([[ 0.75592895,  1.13389342],
       [ 0.37796447,  1.88982237]])
>>> r.dot(r)
array([[ 1.,  3.],
       [ 1.,  4.]])