matrix_norm

matrix_norm(x: array, /, *, keepdims: bool = False, ord: int | float | ~typing.Literal[inf, -inf, 'fro', 'nuc'] | None = 'fro') → array

Computes the matrix norm of a matrix (or a stack of matrices) x.

Parameters:

  • x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a floating-point data type.

  • keepdims (bool) – If True, the last two axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see Broadcasting). Otherwise, if False, the last two axes (dimensions) must not be included in the result. Default: False.

  • ord (Optional[Union[int, float, Literal[inf, -inf, 'fro', 'nuc']]]) –

    order of the norm. The following mathematical norms must be supported:

    ord

    description

    ’fro’

    Frobenius norm

    ’nuc’

    nuclear norm

    1

    max(sum(abs(x), axis=0))

    2

    largest singular value

    inf

    max(sum(abs(x), axis=1))

    The following non-mathematical “norms” must be supported:

    ord

    description

    -1

    min(sum(abs(x), axis=0))

    -2

    smallest singular value

    -inf

    min(sum(abs(x), axis=1))

    If ord=1, the norm corresponds to the induced matrix norm where p=1 (i.e., the maximum absolute value column sum).

    If ord=2, the norm corresponds to the induced matrix norm where p=inf (i.e., the maximum absolute value row sum).

    If ord=inf, the norm corresponds to the induced matrix norm where p=2 (i.e., the largest singular value).

    Default: 'fro'.

Returns:

out (array) – an array containing the norms for each MxN matrix. If keepdims is False, the returned array must have a rank which is two less than the rank of x. If x has a real-valued data type, the returned array must have a real-valued floating-point data type determined by Type Promotion Rules. If x has a complex-valued data type, the returned array must have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then the returned array must have a float64 data type).

Notes

Changed in version 2022.12: Added complex data type support.