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Diffstat (limited to 'numpy/linalg/linalg.py')
| -rw-r--r-- | numpy/linalg/linalg.py | 35 |
1 files changed, 26 insertions, 9 deletions
diff --git a/numpy/linalg/linalg.py b/numpy/linalg/linalg.py index 437db6d2b..f762cf72d 100644 --- a/numpy/linalg/linalg.py +++ b/numpy/linalg/linalg.py @@ -361,30 +361,36 @@ def cholesky(a): """ Cholesky decomposition. - Return the Cholesky decomposition, :math:`A = L L^*` of a Hermitian - positive-definite matrix :math:`A`. + 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 : array_like, shape (M, M) - Hermitian (symmetric, if it is real) and positive definite + Hermitian (symmetric if all elements are real), positive-definite input matrix. Returns ------- - L : array_like, shape (M, M) - Lower-triangular Cholesky factor of A. + L : ndarray, or matrix object if `a` is, shape (M, M) + Lower-triangular Cholesky factor of a. Raises ------ LinAlgError - If the decomposition fails. + If the decomposition fails, for example, if `a` is not + positive-definite. Notes ----- The Cholesky decomposition is often used as a fast way of solving - .. math:: A \\mathbf{x} = \\mathbf{b}. + .. math:: A \\mathbf{x} = \\mathbf{b} + + (when `A` is both Hermitian/symmetric and positive-definite). First, we solve for :math:`\\mathbf{y}` in @@ -392,18 +398,29 @@ def cholesky(a): and then for :math:`\\mathbf{x}` in - .. math:: L^* \\mathbf{x} = \\mathbf{y}. + .. math:: L.H \\mathbf{x} = \\mathbf{y}. 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()) + >>> 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]]) """ a, wrap = _makearray(a) |