numpy.linalg.svd — NumPy v1.15 Manual (original) (raw)

Singular Value Decomposition.

When a is a 2D array, it is factorized as u @ np.diag(s) @ vh = (u * s) @ vh, where u and vh are 2D unitary arrays and s is a 1D array of _a_’s singular values. When a is higher-dimensional, SVD is applied in stacked mode as explained below.

Parameters:	a : (…, M, N) array_like A real or complex array with a.ndim >= 2. full_matrices : bool, optional If True (default), u and vh have the shapes (..., M, M) and(..., N, N), respectively. Otherwise, the shapes are(..., M, K) and (..., K, N), respectively, whereK = min(M, N). compute_uv : bool, optional Whether or not to compute u and vh in addition to s. True by default.
Returns:	u : { (…, M, M), (…, M, K) } array Unitary array(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when_compute_uv_ is True. s : (…, K) array Vector(s) with the singular values, within each vector sorted in descending order. The first a.ndim - 2 dimensions have the same size as those of the input a. vh : { (…, N, N), (…, K, N) } array Unitary array(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when_compute_uv_ is True.
Raises:	LinAlgError If SVD computation does not converge.

Parameters:

a : (…, M, N) array_like A real or complex array with a.ndim >= 2. full_matrices : bool, optional If True (default), u and vh have the shapes (..., M, M) and(..., N, N), respectively. Otherwise, the shapes are(..., M, K) and (..., K, N), respectively, whereK = min(M, N). compute_uv : bool, optional Whether or not to compute u and vh in addition to s. True by default.

Returns:

u : { (…, M, M), (…, M, K) } array Unitary array(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when_compute_uv_ is True. s : (…, K) array Vector(s) with the singular values, within each vector sorted in descending order. The first a.ndim - 2 dimensions have the same size as those of the input a. vh : { (…, N, N), (…, K, N) } array Unitary array(s). The first a.ndim - 2 dimensions have the same size as those of the input a. The size of the last two dimensions depends on the value of full_matrices. Only returned when_compute_uv_ is True.

Raises:

LinAlgError If SVD computation does not converge.

Notes

Changed in version 1.8.0: Broadcasting rules apply, see the numpy.linalg documentation for details.

The decomposition is performed using LAPACK routine _gesdd.

SVD is usually described for the factorization of a 2D matrix A. The higher-dimensional case will be discussed below. In the 2D case, SVD is written as A = U S V^H, where A = a, U= u,S= \mathtt{np.diag}(s) and V^H = vh. The 1D array _s_contains the singular values of a and u and vh are unitary. The rows of vh are the eigenvectors of A^H A and the columns of u are the eigenvectors of A A^H. In both cases the corresponding (possibly non-zero) eigenvalues are given by s**2.

If a has more than two dimensions, then broadcasting rules apply, as explained in Linear algebra on several matrices at once. This means that SVD is working in “stacked” mode: it iterates over all indices of the firsta.ndim - 2 dimensions and for each combination SVD is applied to the last two indices. The matrix a can be reconstructed from the decomposition with either (u * s[..., None, :]) @ vh oru @ (s[..., None] * vh). (The @ operator can be replaced by the function np.matmul for python versions below 3.5.)

If a is a matrix object (as opposed to an ndarray), then so are all the return values.

Examples

a = np.random.randn(9, 6) + 1jnp.random.randn(9, 6) b = np.random.randn(2, 7, 8, 3) + 1jnp.random.randn(2, 7, 8, 3)

Reconstruction based on full SVD, 2D case:

u, s, vh = np.linalg.svd(a, full_matrices=True) u.shape, s.shape, vh.shape ((9, 9), (6,), (6, 6)) np.allclose(a, np.dot(u[:, :6] * s, vh)) True smat = np.zeros((9, 6), dtype=complex) smat[:6, :6] = np.diag(s) np.allclose(a, np.dot(u, np.dot(smat, vh))) True

Reconstruction based on reduced SVD, 2D case:

u, s, vh = np.linalg.svd(a, full_matrices=False) u.shape, s.shape, vh.shape ((9, 6), (6,), (6, 6)) np.allclose(a, np.dot(u * s, vh)) True smat = np.diag(s) np.allclose(a, np.dot(u, np.dot(smat, vh))) True

Reconstruction based on full SVD, 4D case:

u, s, vh = np.linalg.svd(b, full_matrices=True) u.shape, s.shape, vh.shape ((2, 7, 8, 8), (2, 7, 3), (2, 7, 3, 3)) np.allclose(b, np.matmul(u[..., :3] * s[..., None, :], vh)) True np.allclose(b, np.matmul(u[..., :3], s[..., None] * vh)) True

Reconstruction based on reduced SVD, 4D case:

u, s, vh = np.linalg.svd(b, full_matrices=False) u.shape, s.shape, vh.shape ((2, 7, 8, 3), (2, 7, 3), (2, 7, 3, 3)) np.allclose(b, np.matmul(u * s[..., None, :], vh)) True np.allclose(b, np.matmul(u, s[..., None] * vh)) True