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jax.numpy.linalg.svd — JAX documentation

jax.numpy.linalg.svd(a, full_matrices=True, compute_uv=True, hermitian=False, subset_by_index=None)[source]#

Compute the singular value decomposition.

JAX implementation of numpy.linalg.svd(), implemented in terms of jax.lax.linalg.svd().

The SVD of a matrix A is given by

\[A = U\Sigma V^H\]

• \(U\) contains the left singular vectors and satisfies \(U^HU=I\)

• \(V\) contains the right singular vectors and satisfies \(V^HV=I\)

• \(\Sigma\) is a diagonal matrix of singular values.

Parameters:

• a (ArrayLike) – input array, of shape (..., N, M)

• full_matrices (bool) – if True (default) compute the full matrices; i.e. u and vh have shape (..., N, N) and (..., M, M). If False, then the shapes are (..., N, K) and (..., K, M) with K = min(N, M).

• compute_uv (bool) – if True (default), return the full SVD (u, s, vh). If False then return only the singular values s.

• hermitian (bool) – if True, assume the matrix is hermitian, which allows for a more efficient implementation (default=False)

• subset_by_index (tuple[int, int] | None) – (TPU-only) Optional 2-tuple [start, end] indicating the range of indices of singular values to compute. For example, if [n-2, n] then svd computes the two largest singular values and their singular vectors. Only compatible with full_matrices=False.

Returns:

A tuple of arrays (u, s, vh) if compute_uv is True, otherwise the array s.

• u: left singular vectors of shape (..., N, N) if full_matrices is True or (..., N, K) otherwise.

• s: singular values of shape (..., K)

• vh: conjugate-transposed right singular vectors of shape (..., M, M) if full_matrices is True or (..., K, M) otherwise.

where K = min(N, M).

Return type:

Array | SVDResult

Examples

Consider the SVD of a small real-valued array:

>>> x = jnp.array([[1., 2., 3.],
...                [6., 5., 4.]])
>>> u, s, vt = jnp.linalg.svd(x, full_matrices=False)
>>> s  
Array([9.361919 , 1.8315067], dtype=float32)

The singular vectors are in the columns of u and v = vt.T. These vectors are orthonormal, which can be demonstrated by comparing the matrix product with the identity matrix:

>>> jnp.allclose(u.T @ u, jnp.eye(2), atol=1E-5)
Array(True, dtype=bool)
>>> v = vt.T
>>> jnp.allclose(v.T @ v, jnp.eye(2), atol=1E-5)
Array(True, dtype=bool)

Given the SVD, x can be reconstructed via matrix multiplication:

>>> x_reconstructed = u @ jnp.diag(s) @ vt
>>> jnp.allclose(x_reconstructed, x)
Array(True, dtype=bool)

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