# 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.

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”. 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] (1, 2) R. A. Horn and C. R. Johnson, “Matrix Analysis”, Cambridge University Press, 1985.
 [2] (1, 2) N. J. Higham, “Functions of Matrices: Theory and Computation”, SIAM, 2008.

Examples

>>> 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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