toqito.matrix_props.is_projection
=================================

.. py:module:: toqito.matrix_props.is_projection

.. autoapi-nested-parse::

   Checks if the matrix is a projection matrix.





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

.. py:function:: is_projection(mat, rtol = 1e-05, atol = 1e-08)

   Check if matrix is a projection matrix [@WikiProjMat].

   A matrix is a projection matrix if it is positive semidefinite (PSD) and if

   \[
       \begin{equation}
           X^2 = X
       \end{equation}
   \]

   where \(X\) is the matrix in question.

   .. rubric:: Examples

   Consider the following matrix

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

   our function indicates that this is indeed a projection matrix.

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

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

   print(is_projection(A))
   ```

   Alternatively, the following example matrix \(B\) defined as

   \[
       B = \begin{pmatrix}
               -1 & -1 \\
               -1 & -1
           \end{pmatrix}
   \]

   is not positive definite.

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

   B = np.array([[-1, -1], [-1, -1]])

   print(is_projection(B))
   ```

   :param mat: Matrix to check.
   :param rtol: The relative tolerance parameter (default 1e-05).
   :param atol: The absolute tolerance parameter (default 1e-08).

   :returns: Return `True` if matrix is a projection matrix, and `False` otherwise.