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# numpy.linalg.cholesky

`numpy.linalg.``cholesky`(a)

  Cholesky decomposition.

  Return the Cholesky decomposition, L * L.H, of the square matrix a, where L is lower-triangular and .H is the conjugate transpose operator (which is the ordinary transpose if a is real-valued). a must be Hermitian (symmetric if real-valued) and positive-definite. Only L is actually returned.

  Parameters: | a : (..., M, M) array_like Hermitian (symmetric if all elements are real), positive-definite input matrix.
  Returns: | L : (..., M, M) array_like Upper or lower-triangular Cholesky factor of a. Returns a matrix object if a is a matrix object.
  Raises: | LinAlgError : If the decomposition fails, for example, if a is not positive-definite.

  Notes

  Broadcasting rules apply, see the `numpy.linalg` documentation for details.

  The Cholesky decomposition is often used as a fast way of solving

  (when A is both Hermitian/symmetric and positive-definite).

  First, we solve for in

  and then for in

  Examples

      >>> A = np.array([[1,-2j],[2j,5]])
      >>> A
      array([[ 1.+0.j,  0.-2.j],
             [ 0.+2.j,  5.+0.j]])
      >>> L = np.linalg.cholesky(A)
      >>> L
      array([[ 1.+0.j,  0.+0.j],
             [ 0.+2.j,  1.+0.j]])
      >>> np.dot(L, L.T.conj()) # verify that L * L.H = A
      array([[ 1.+0.j,  0.-2.j],
             [ 0.+2.j,  5.+0.j]])
      >>> A = [[1,-2j],[2j,5]] # what happens if A is only array_like?
      >>> np.linalg.cholesky(A) # an ndarray object is returned
      array([[ 1.+0.j,  0.+0.j],
             [ 0.+2.j,  1.+0.j]])
      >>> # But a matrix object is returned if A is a matrix object
      >>> LA.cholesky(np.matrix(A))
      matrix([[ 1.+0.j,  0.+0.j],
              [ 0.+2.j,  1.+0.j]])