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is_anti_hermitian

Checks if the matrix is an anti-Hermitian matrix.

is_anti_hermitian 

is_anti_hermitian(
    mat: ndarray, rtol: float = 1e-05, atol: float = 1e-08
) -> bool

Check if matrix is anti-Hermitian (a.k.a. skew-Hermitian) 1.

An anti-Hermitian matrix is a complex square matrix that is equal to the negative of its own conjugate transpose.

Parameters:

  • mat (ndarray) –

    Matrix to check.

  • 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 matrix is anti-Hermitian, and False otherwise.

Examples:

Consider the following matrix:

\[ A = \begin{pmatrix} 2j & -1 + 2j & 4j \\ 1 + 2j & 3j & -1 \\ 4j & 1 & 1j \end{pmatrix} \]

our function indicates that this is indeed an anti-Hermitian matrix as it holds that

[ A = -A^*. ] 

import numpy as np
from toqito.matrix_props import is_anti_hermitian

mat = np.array([[2j, -1 + 2j, 4j], [1 + 2j, 3j, -1], [4j, 1, 1j]])

print(is_anti_hermitian(mat))

True

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

\[ B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \]

is not anti-Hermitian. 

import numpy as np
from toqito.matrix_props import is_anti_hermitian

mat = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

print(is_anti_hermitian(mat))

False

References

1 Wikipedia. Skew-Hermitian matrix. link.

Source code in toqito/matrix_props/is_anti_hermitian.py 
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def is_anti_hermitian(mat: np.ndarray, rtol: float = 1e-05, atol: float = 1e-08) -> bool:
    r"""Check if matrix is anti-Hermitian (a.k.a. skew-Hermitian) [@wikipediaskewhermitian].

    An anti-Hermitian matrix is a complex square matrix that is equal to the negative of its own
    conjugate transpose.

    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 anti-Hermitian, and False otherwise.

    Examples:
        Consider the following matrix:

        \[
            A = \begin{pmatrix}
                    2j & -1 + 2j & 4j \\
                    1 + 2j & 3j & -1 \\
                    4j & 1 & 1j
                \end{pmatrix}
        \]

        our function indicates that this is indeed an anti-Hermitian matrix as it holds that

        \[
            A = -A^*.
        \]
        ```python exec="1" source="above" result="text"
        import numpy as np
        from toqito.matrix_props import is_anti_hermitian

        mat = np.array([[2j, -1 + 2j, 4j], [1 + 2j, 3j, -1], [4j, 1, 1j]])

        print(is_anti_hermitian(mat))
        ```

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

        \[
            B = \begin{pmatrix}
                    1 & 2 & 3 \\
                    4 & 5 & 6 \\
                    7 & 8 & 9
                \end{pmatrix}
        \]

        is not anti-Hermitian.
        ```python exec="1" source="above" result="text"
        import numpy as np
        from toqito.matrix_props import is_anti_hermitian

        mat = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

        print(is_anti_hermitian(mat))
        ```

    """
    return is_hermitian(mat * 1j, rtol, atol)