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”) or a = 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.

The documentation is written assuming array arguments are of specified “core” shapes. However, 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:
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”, then a = 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]])
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