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is_diagonally_dominant

Checks if the matrix is diagonally dominant.

is_diagonally_dominant 

is_diagonally_dominant(
    mat: ndarray, is_strict: bool = True
) -> bool

Check if matrix is diagonal dominant (DD) 1.

A matrix is diagonally dominant if the matrix is square and if for every row of the matrix, the magnitude of the diagonal entry in a row is greater than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row.

Parameters:

  • mat (ndarray) –

    Matrix to check.

  • is_strict (bool, default: True ) –

    Whether the inequality is strict.

Returns:

  • bool –

    Return True if matrix is diagonally dominant, and False otherwise.

Examples:

The following is an example of a 3-by-3 diagonal matrix:

\[ A = \begin{pmatrix} 2 & -1 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix} \]

our function indicates that this is indeed a diagonally dominant matrix.

import numpy as np
from toqito.matrix_props import is_diagonally_dominant

A = np.array([[2, -1, 0], [0, 2, -1], [0, -1, 2]])

print(is_diagonally_dominant(A))

True

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

\[ B = \begin{pmatrix} -1 & 2 \\ -1 & -1 \end{pmatrix} \]

is not diagonally dominant.

import numpy as np
from toqito.matrix_props import is_diagonally_dominant

B = np.array([[-1, 2], [-1, -1]])

print(is_diagonally_dominant(B))

False

References

1 Wikipedia. Diagonally dominant matrix. link.

Source code in toqito/matrix_props/is_diagonally_dominant.py 
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def is_diagonally_dominant(mat: np.ndarray, is_strict: bool = True) -> bool:
    r"""Check if matrix is diagonal dominant (DD) [@wikipediadiagonallydominant].

    A matrix is diagonally dominant if the matrix is square
    and if for every row of the matrix, the magnitude of the diagonal entry in a row is greater
    than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row.

    Args:
        mat: Matrix to check.
        is_strict: Whether the inequality is strict.

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

    Examples:
        The following is an example of a 3-by-3 diagonal matrix:

        \[
            A = \begin{pmatrix}
                    2 & -1 & 0 \\
                    0 & 2 & -1 \\
                    0 & -1 & 2
                \end{pmatrix}
        \]

        our function indicates that this is indeed a diagonally dominant matrix.

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

        A = np.array([[2, -1, 0], [0, 2, -1], [0, -1, 2]])

        print(is_diagonally_dominant(A))
        ```

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

        \[
            B = \begin{pmatrix}
                    -1 & 2 \\
                    -1 & -1
                \end{pmatrix}
        \]

        is not diagonally dominant.

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

        B = np.array([[-1, 2], [-1, -1]])

        print(is_diagonally_dominant(B))
        ```

    """
    if not is_square(mat):
        return False

    mat = np.abs(mat)
    diag_coeffs = np.diag(mat)
    row_sum = np.sum(mat, axis=1) - diag_coeffs
    return bool(np.all(diag_coeffs > row_sum) if is_strict else np.all(diag_coeffs >= row_sum))