scipy.linalg.norm¶

scipy.linalg.
norm
(a, ord=None, axis=None, keepdims=False)[source]¶ Matrix or vector norm.
This function is able to return one of seven different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the
ord
parameter. Parameters
 a(M,) or (M, N) array_like
Input array. If axis is None, a must be 1D or 2D.
 ord{nonzero int, inf, inf, ‘fro’}, optional
Order of the norm (see table under
Notes
). inf means numpy’s inf object axis{int, 2tuple of ints, None}, optional
If axis is an integer, it specifies the axis of a along which to compute the vector norms. If axis is a 2tuple, it specifies the axes that hold 2D matrices, and the matrix norms of these matrices are computed. If axis is None then either a vector norm (when a is 1D) or a matrix norm (when a is 2D) is returned.
 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 a.
 Returns
 nfloat or ndarray
Norm of the matrix or vector(s).
Notes
For values of
ord <= 0
, 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
2norm
‘fro’
Frobenius 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
2norm (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
axis
andkeepdims
arguments are passed directly tonumpy.linalg.norm
and are only usable if they are supported by the version of numpy in use.References
 1(1,2)
G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples
>>> from scipy.linalg import norm >>> a = np.arange(9)  4.0 >>> a array([4., 3., 2., 1., 0., 1., 2., 3., 4.]) >>> b = a.reshape((3, 3)) >>> b array([[4., 3., 2.], [1., 0., 1.], [ 2., 3., 4.]])
>>> norm(a) 7.745966692414834 >>> norm(b) 7.745966692414834 >>> norm(b, 'fro') 7.745966692414834 >>> norm(a, np.inf) 4 >>> norm(b, np.inf) 9 >>> norm(a, np.inf) 0 >>> norm(b, np.inf) 2
>>> norm(a, 1) 20 >>> norm(b, 1) 7 >>> norm(a, 1) 4.6566128774142013e010 >>> norm(b, 1) 6 >>> norm(a, 2) 7.745966692414834 >>> norm(b, 2) 7.3484692283495345
>>> norm(a, 2) 0 >>> norm(b, 2) 1.8570331885190563e016 >>> norm(a, 3) 5.8480354764257312 >>> norm(a, 3) 0