cr.sparse.la.null_space

cr.sparse.la.null_space(A, rcond=None)[source]

Constructs an orthonormal basis for the null space of A using SVD

Parameters

  • A (jax.numpy.ndarray) – Input matrix of size (M, N) where M is the dimension of the ambient vector space and N is the number of vectors in A

  • rcond (float) – Relative condition number. Singular values s smaller than rcond * max(s) are considered zero. Default: floating point eps * max(M,N).

Returns

Returns a tuple consisting of

  • the right singular vectors of A

  • the effective rank of A

Return type

(jax.numpy.ndarray, int)

To get the ONB for the null space of A, follow the two step process:

Z, r = null_space(A)
Z = Z[:, r:]

The dimension of the effective null space is \(N - r\) where r is the rank of A.

Examples

>>> A = random.normal(key, (3, 5))
>>> Z, r = null_space(A)
>>> Z = Z[:, r:]
>>> Z.shape
(5, 2)
>>> print(jnp.allclose(A @ Z, 0))
True