toqito.matrix_ops.null_space
============================

.. py:module:: toqito.matrix_ops.null_space

.. autoapi-nested-parse::

   Compute an orthonormal basis for the null space of a matrix.





Module Contents
---------------

.. py:function:: null_space(mat, tol = 1e-08)

   Return an orthonormal basis for the kernel of ``mat`` [@WikiNullSpace].

   The routine employs the singular value decomposition so that the columns of the
   returned matrix span the null space and are orthonormal with respect to the
   standard inner product.

   .. rubric:: Examples

   Consider the matrix

   \[
       A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}.
   \]

   Its null space is spanned by the vectors \((1,-1,0)\) and \((0,0,1)\).

   ```python exec="1" source="above"
   import numpy as np
   from toqito.matrix_ops import null_space

   A = np.array([[1, 1, 0], [0, 0, 0]], dtype=float)
   null_basis = null_space(A)
   print(null_basis)
   ```

   :param mat: Matrix whose null space is sought.
   :param tol: Numerical tolerance that distinguishes zero singular values.

   :returns: A matrix whose columns form an orthonormal basis for the null space.