scipy.linalg.polar#
- scipy.linalg.polar(a, side='right')[source]#
Compute the polar decomposition.
Returns the factors of the polar decomposition [1] u and p such that
a = up
(if side is “right”) ora = pu
(if side is “left”), where p is positive semidefinite. Depending on the shape of a, either the rows or columns of u are orthonormal. When a is a square array, u is a square unitary array. When a is not square, the “canonical polar decomposition” [2] is computed.- Parameters:
- a(m, n) array_like
The array to be factored.
- side{‘left’, ‘right’}, optional
Determines whether a right or left polar decomposition is computed. If side is “right”, then
a = up
. If side is “left”, thena = pu
. The default is “right”.
- Returns:
- u(m, n) ndarray
If a is square, then u is unitary. If m > n, then the columns of a are orthonormal, and if m < n, then the rows of u are orthonormal.
- pndarray
p is Hermitian positive semidefinite. If a is nonsingular, p is positive definite. The shape of p is (n, n) or (m, m), depending on whether side is “right” or “left”, respectively.
References
[1]R. A. Horn and C. R. Johnson, “Matrix Analysis”, Cambridge University Press, 1985.
[2]N. J. Higham, “Functions of Matrices: Theory and Computation”, SIAM, 2008.
Examples
>>> import numpy as np >>> from scipy.linalg import polar >>> a = np.array([[1, -1], [2, 4]]) >>> u, p = polar(a) >>> u array([[ 0.85749293, -0.51449576], [ 0.51449576, 0.85749293]]) >>> p array([[ 1.88648444, 1.2004901 ], [ 1.2004901 , 3.94446746]])
A non-square example, with m < n:
>>> b = np.array([[0.5, 1, 2], [1.5, 3, 4]]) >>> u, p = polar(b) >>> u array([[-0.21196618, -0.42393237, 0.88054056], [ 0.39378971, 0.78757942, 0.4739708 ]]) >>> p array([[ 0.48470147, 0.96940295, 1.15122648], [ 0.96940295, 1.9388059 , 2.30245295], [ 1.15122648, 2.30245295, 3.65696431]]) >>> u.dot(p) # Verify the decomposition. array([[ 0.5, 1. , 2. ], [ 1.5, 3. , 4. ]]) >>> u.dot(u.T) # The rows of u are orthonormal. array([[ 1.00000000e+00, -2.07353665e-17], [ -2.07353665e-17, 1.00000000e+00]])
Another non-square example, with m > n:
>>> c = b.T >>> u, p = polar(c) >>> u array([[-0.21196618, 0.39378971], [-0.42393237, 0.78757942], [ 0.88054056, 0.4739708 ]]) >>> p array([[ 1.23116567, 1.93241587], [ 1.93241587, 4.84930602]]) >>> u.dot(p) # Verify the decomposition. array([[ 0.5, 1.5], [ 1. , 3. ], [ 2. , 4. ]]) >>> u.T.dot(u) # The columns of u are orthonormal. array([[ 1.00000000e+00, -1.26363763e-16], [ -1.26363763e-16, 1.00000000e+00]])