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is_pseudo_unitary

Checks if matrix is pseudo unitary.

is_pseudo_unitary 

is_pseudo_unitary(
    mat: ndarray,
    p: int,
    q: int,
    rtol: float = 1e-05,
    atol: float = 1e-08,
) -> bool

Check if a matrix is pseudo-unitary.

A matrix A of size (p+q)x(p+q) is pseudo-unitary with respect to a given signature matrix J if it satisfies

\[ A^* J A = J, \]

where:

  • \(A^*\) is the conjugate transpose (Hermitian transpose) of \(A\),
  • \(J\) is a diagonal matrix with first \(p\) diagonal entries equal to 1 and next \(q\) diagonal entries equal to -1.

Parameters:

  • mat (ndarray) –

    The matrix to check.

  • p (int) –

    Number of positive entries in the signature matrix.

  • q (int) –

    Number of negative entries in the signature matrix.

  • rtol (float, default: 1e-05 ) –

    The relative tolerance parameter (default 1e-05).

  • atol (float, default: 1e-08 ) –

    The absolute tolerance parameter (default 1e-08).

Returns:

  • bool –

    Return True if the matrix is pseudo-unitary, and False otherwise.

Raises:

  • ValueError –

    When p < 0 or q < 0.

Examples:

Consider the following matrix:

\[ A = \begin{pmatrix} \cosh(1) & \sinh(1) \\ \sinh(1) & \cosh(1) \end{pmatrix} \]

with the signature matrix:

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

Our function confirms that \(A\) is pseudo-unitary.

import numpy as np
from toqito.matrix_props import is_pseudo_unitary

A = np.array([[np.cosh(1), np.sinh(1)], [np.sinh(1), np.cosh(1)]])

print(is_pseudo_unitary(A, p=1, q=1))

True

Alternatively, the following matrix \(B\)

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

is not pseudo-unitary with respect to the same signature matrix:

import numpy as np
from toqito.matrix_props import is_pseudo_unitary

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

print(is_pseudo_unitary(B, p=1, q=1))

False
Source code in toqito/matrix_props/is_pseudo_unitary.py 
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def is_pseudo_unitary(mat: np.ndarray, p: int, q: int, rtol: float = 1e-05, atol: float = 1e-08) -> bool:
    r"""Check if a matrix is pseudo-unitary.

    A matrix A of size (p+q)x(p+q) is pseudo-unitary with respect to a given signature matrix J if it satisfies

    \[
        A^* J A = J,
    \]

    where:

    - \(A^*\) is the conjugate transpose (Hermitian transpose) of \(A\),
    - \(J\) is a diagonal matrix with first \(p\) diagonal entries equal to 1 and next \(q\)
        diagonal entries equal to -1.

    Args:
        mat: The matrix to check.
        p: Number of positive entries in the signature matrix.
        q: Number of negative entries in the signature matrix.
        rtol: The relative tolerance parameter (default 1e-05).
        atol: The absolute tolerance parameter (default 1e-08).

    Returns:
        Return `True` if the matrix is pseudo-unitary, and `False` otherwise.

    Raises:
        ValueError: When p < 0 or q < 0.

    Examples:
        Consider the following matrix:

        \[
            A = \begin{pmatrix}
                \cosh(1) & \sinh(1) \\
                \sinh(1) & \cosh(1)
            \end{pmatrix}
        \]

        with the signature matrix:

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

        Our function confirms that \(A\) is pseudo-unitary.

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

        A = np.array([[np.cosh(1), np.sinh(1)], [np.sinh(1), np.cosh(1)]])

        print(is_pseudo_unitary(A, p=1, q=1))
        ```


        Alternatively, the following matrix \(B\)

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

        is not pseudo-unitary with respect to the same signature matrix:

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

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

        print(is_pseudo_unitary(B, p=1, q=1))
        ```

    """
    if p < 0 or q < 0:
        raise ValueError("p and q must be non-negative")

    if not is_square(mat):
        return False

    if p + q != mat.shape[0]:
        return False

    signature = np.diag(np.hstack((np.ones(p), -np.ones(q))))
    ac_j_a_mat = mat.conj().T @ signature @ mat

    return bool(np.allclose(ac_j_a_mat, signature, rtol=rtol, atol=atol))