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# scipy.linalg.polar

scipy.linalg.polar(a, side='right')

  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]])