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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.
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.
p : ndarray
`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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