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is_orthonormal

Checks if the set of vectors are orthonormal.

is_orthonormal 

is_orthonormal(vectors: list[ndarray]) -> bool

Check if the vectors are orthonormal.

Parameters:

  • vectors (list[ndarray]) –

    A list of np.ndarray vectors, each given as a 1-D array or a column/row vector.

Returns:

  • bool –

    True if vectors are orthonormal; False otherwise.

Examples:

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

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

To check these are a known set of orthonormal vectors:

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 = [v_1, v_2, v_3]

print(is_orthonormal(v))

True
Source code in toqito/matrix_props/is_orthonormal.py 
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def is_orthonormal(vectors: list[np.ndarray]) -> bool:
    r"""Check if the vectors are orthonormal.

    Args:
        vectors: A list of `np.ndarray` vectors, each given as a 1-D array or a column/row vector.

    Returns:
        True if vectors are orthonormal; False otherwise.

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

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

        To check these are a known set of orthonormal vectors:

        ```python exec="1" source="above" result="text"
        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 = [v_1, v_2, v_3]

        print(is_orthonormal(v))
        ```

    """
    # Flatten each vector to 1-D so column vectors (n, 1) and row vectors (1, n) are handled the
    # same as plain 1-D arrays; the Gram matrix of an orthonormal set is the identity.
    mat = np.array([np.asarray(vec).reshape(-1) for vec in vectors])
    return np.allclose(mat @ mat.conj().T, np.eye(len(vectors)))