Source code for toqito.matrix_props.is_orthonormal

"""Checks if the set of vectors are orthonormal."""

import numpy as np

from toqito.state_props.is_mutually_orthogonal import is_mutually_orthogonal




def is_orthonormal(vectors: list[np.ndarray]) -> bool:
    r"""Check if the vectors are orthonormal.

    Examples:
        The following vectors are an example of an orthonormal set of
        vectors in \(\mathbb{R}^3\).

        \[
            \begin{pmatrix}
                1 \\ 0 \\ 1
            \end{pmatrix}, \quad
            \begin{pmatrix}
                1 \\ 1 \\ 0
            \end{pmatrix}, \quad \text{and} \quad
            \begin{pmatrix}
                0 \\ 0 \\ 1
            \end{pmatrix}
        \]

        To check these are a known set of orthonormal vectors:

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

        v_1 = np.array([1, 0, 0])
        v_2 = np.array([0, 1, 0])
        v_3 = np.array([0, 0, 1])

        v = np.array([v_1, v_2, v_3])

        print(is_orthonormal(v))
        ```


    Args:
        vectors: A list of `np.ndarray` 1-by-n vectors.

    Returns:
        True if vectors are orthonormal; False otherwise.

    """
    return is_mutually_orthogonal(vectors) and np.allclose(
        np.dot(vectors, np.conjugate(vectors).T), np.eye(vectors.shape[0])
    )