toqito.matrix_props.is_positive_definite
========================================

.. py:module:: toqito.matrix_props.is_positive_definite

.. autoapi-nested-parse::

   Checks if the matrix is a positive definite matrix.





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

.. py:function:: is_positive_definite(mat)

   Check if matrix is positive definite (PD) [@WikiPosDef].

   .. rubric:: Examples

   Consider the following matrix

   \[
       A = \begin{pmatrix}
               2 & -1 & 0 \\
               -1 & 2 & -1 \\
               0 & -1 & 2
           \end{pmatrix}
   \]

   our function indicates that this is indeed a positive definite matrix.

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

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

   print(is_positive_definite(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_positive_definite

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

   print(is_positive_definite(B))
   ```

   !!! See Also
       [`is_positive_semidefinite`][toqito.matrix_props.is_positive_semidefinite.is_positive_semidefinite]

   :param mat: Matrix to check.

   :returns: Return `True` if matrix is positive definite, and `False` otherwise.