toqito.matrix_props.spark
=========================

.. py:module:: toqito.matrix_props.spark

.. autoapi-nested-parse::

   Computes the spark of a matrix.





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

.. py:function:: spark(mat)

   Compute the spark of a matrix.

   The spark of a matrix A is the smallest number of columns from A that are linearly
   dependent [@Elad_2010_Sparse].

   .. rubric:: Examples

   ```python exec="1" source="above"
   import numpy as np
   from toqito.matrix_props import spark

   A = np.array([[1, 0, 1, 2],
                 [0, 1, 1, 3],
                 [1, 1, 2, 5]])

   print(spark(A))
   ```

   .. rubric:: Notes

   - This function only works for 2D NumPy arrays.
   - If all columns are linearly independent, the function returns n_cols + 1.
   - The time complexity of this implementation is O(2^n) in the worst case,
     where n is the number of columns.
   - For an m x n matrix A with n >= m:
       - If spark(A) = m + 1, then rank(A) = m (full rank).
       - If spark(A) = 1, then the matrix has a zero column.
       - If spark(A) <= rank(A) + 1, then the matrix has dependent columns.

   :raises ValueError: If the input is not a 2D NumPy array.

   :param mat: The input matrix as a 2D NumPy array.

   :returns: The spark of the input matrix `mat`.