Source code for toqito.matrix_props.is_positive_semidefinite

"""Checks if the matrix is a positive semidefinite matrix."""

import numpy as np

from toqito.matrix_props import is_hermitian




def is_positive_semidefinite(mat: np.ndarray, rtol: float = 1e-05, atol: float = 1e-08) -> bool:
    r"""Check if matrix is positive semidefinite (PSD) [@WikiPosDef].

    Examples:
        Consider the following matrix

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

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

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

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

        print(is_positive_semidefinite(A))
        ```

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

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

        is not positive semidefinite.

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

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

        print(is_positive_semidefinite(B))
        ```

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

    Returns:
        Return `True` if matrix is PSD, and `False` otherwise.

    """
    if not is_hermitian(mat, rtol, atol):
        return False
    evals, _ = np.linalg.eigh(mat)
    return all(x >= -abs(atol) for x in evals)