numpy.linalg.norm

numpy.linalg.norm(x, ord=None, axis=None, keepdims=False)[source]

Matrix or vector norm.

This function is able to return one of eight different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ord parameter.

Parameters
xarray_like

Input array. If axis is None, x must be 1-D or 2-D, unless ord is None. If both axis and ord are None, the 2-norm of x.ravel will be returned.

ord{non-zero int, inf, -inf, ‘fro’, ‘nuc’}, optional

Order of the norm (see table under Notes). inf means numpy’s inf object. The default is None.

axis{None, int, 2-tuple of ints}, optional.

If axis is an integer, it specifies the axis of x along which to compute the vector norms. If axis is a 2-tuple, it specifies the axes that hold 2-D matrices, and the matrix norms of these matrices are computed. If axis is None then either a vector norm (when x is 1-D) or a matrix norm (when x is 2-D) is returned. The default is None.

New in version 1.8.0.

keepdimsbool, optional

If this is set to True, the axes which are normed over are left in the result as dimensions with size one. With this option the result will broadcast correctly against the original x.

New in version 1.10.0.

Returns
nfloat or ndarray

Norm of the matrix or vector(s).

See also

scipy.linalg.norm

Similar function in SciPy.

Notes

For values of ord < 1, the result is, strictly speaking, not a mathematical ‘norm’, but it may still be useful for various numerical purposes.

The following norms can be calculated:

ord

norm for matrices

norm for vectors

None

Frobenius norm

2-norm

‘fro’

Frobenius norm

‘nuc’

nuclear norm

inf

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

max(abs(x))

-inf

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

min(abs(x))

0

sum(x != 0)

1

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

as below

-1

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

as below

2

2-norm (largest sing. value)

as below

-2

smallest singular value

as below

other

sum(abs(x)**ord)**(1./ord)

The Frobenius norm is given by [1]:

||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}

The nuclear norm is the sum of the singular values.

Both the Frobenius and nuclear norm orders are only defined for matrices and raise a ValueError when x.ndim != 2.

References

1

G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

Examples

>>> from numpy import linalg as LA
>>> a = np.arange(9) - 4
>>> a
array([-4, -3, -2, ...,  2,  3,  4])
>>> b = a.reshape((3, 3))
>>> b
array([[-4, -3, -2],
       [-1,  0,  1],
       [ 2,  3,  4]])
>>> LA.norm(a)
7.745966692414834
>>> LA.norm(b)
7.745966692414834
>>> LA.norm(b, 'fro')
7.745966692414834
>>> LA.norm(a, np.inf)
4.0
>>> LA.norm(b, np.inf)
9.0
>>> LA.norm(a, -np.inf)
0.0
>>> LA.norm(b, -np.inf)
2.0
>>> LA.norm(a, 1)
20.0
>>> LA.norm(b, 1)
7.0
>>> LA.norm(a, -1)
-4.6566128774142013e-010
>>> LA.norm(b, -1)
6.0
>>> LA.norm(a, 2)
7.745966692414834
>>> LA.norm(b, 2)
7.3484692283495345
>>> LA.norm(a, -2)
0.0
>>> LA.norm(b, -2)
1.8570331885190563e-016 # may vary
>>> LA.norm(a, 3)
5.8480354764257312 # may vary
>>> LA.norm(a, -3)
0.0

Using the axis argument to compute vector norms:

>>> c = np.array([[ 1, 2, 3],
...               [-1, 1, 4]])
>>> LA.norm(c, axis=0)
array([ 1.41421356,  2.23606798,  5.        ])
>>> LA.norm(c, axis=1)
array([ 3.74165739,  4.24264069])
>>> LA.norm(c, ord=1, axis=1)
array([ 6.,  6.])

Using the axis argument to compute matrix norms:

>>> m = np.arange(8).reshape(2,2,2)
>>> LA.norm(m, axis=(1,2))
array([  3.74165739,  11.22497216])
>>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :])
(3.7416573867739413, 11.224972160321824)