# scipy.linalg.svd#

scipy.linalg.svd(a, full_matrices=True, compute_uv=True, overwrite_a=False, check_finite=True, lapack_driver='gesdd')[source]#

Singular Value Decomposition.

Factorizes the matrix a into two unitary matrices `U` and `Vh`, and a 1-D array `s` of singular values (real, non-negative) such that `a == U @ S @ Vh`, where `S` is a suitably shaped matrix of zeros with main diagonal `s`.

Parameters
a(M, N) array_like

Matrix to decompose.

full_matricesbool, optional

If True (default), U and Vh are of shape `(M, M)`, `(N, N)`. If False, the shapes are `(M, K)` and `(K, N)`, where `K = min(M, N)`.

compute_uvbool, optional

Whether to compute also `U` and `Vh` in addition to `s`. Default is True.

overwrite_abool, optional

Whether to overwrite a; may improve performance. Default is False.

check_finitebool, optional

Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

lapack_driver{‘gesdd’, ‘gesvd’}, optional

Whether to use the more efficient divide-and-conquer approach (`'gesdd'`) or general rectangular approach (`'gesvd'`) to compute the SVD. MATLAB and Octave use the `'gesvd'` approach. Default is `'gesdd'`.

New in version 0.18.

Returns
Undarray

Unitary matrix having left singular vectors as columns. Of shape `(M, M)` or `(M, K)`, depending on full_matrices.

sndarray

The singular values, sorted in non-increasing order. Of shape (K,), with `K = min(M, N)`.

Vhndarray

Unitary matrix having right singular vectors as rows. Of shape `(N, N)` or `(K, N)` depending on full_matrices.

For `compute_uv=False`, only `s` is returned.
Raises
LinAlgError

If SVD computation does not converge.

`svdvals`

Compute singular values of a matrix.

`diagsvd`

Construct the Sigma matrix, given the vector s.

Examples

```>>> from scipy import linalg
>>> from numpy.random import default_rng
>>> rng = default_rng()
>>> m, n = 9, 6
>>> a = rng.standard_normal((m, n)) + 1.j*rng.standard_normal((m, n))
>>> U, s, Vh = linalg.svd(a)
>>> U.shape,  s.shape, Vh.shape
((9, 9), (6,), (6, 6))
```

Reconstruct the original matrix from the decomposition:

```>>> sigma = np.zeros((m, n))
>>> for i in range(min(m, n)):
...     sigma[i, i] = s[i]
>>> a1 = np.dot(U, np.dot(sigma, Vh))
>>> np.allclose(a, a1)
True
```

Alternatively, use `full_matrices=False` (notice that the shape of `U` is then `(m, n)` instead of `(m, m)`):

```>>> U, s, Vh = linalg.svd(a, full_matrices=False)
>>> U.shape, s.shape, Vh.shape
((9, 6), (6,), (6, 6))
>>> S = np.diag(s)
>>> np.allclose(a, np.dot(U, np.dot(S, Vh)))
True
```
```>>> s2 = linalg.svd(a, compute_uv=False)
>>> np.allclose(s, s2)
True
```