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# scipy.linalg.norm

`scipy.linalg.``norm`(a, ord=None, axis=None, keepdims=False)

  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 1-D or 2-D.

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

  > Order of the norm (see table under `Notes`). inf means numpy’s inf object

  axis : {int, 2-tuple 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 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 a is 1-D) or a matrix norm (when a is 2-D) is returned.

  keepdims : bool, 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:

  n : float 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 | 2-norm
  ‘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 | 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 [R93]:

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

  The `axis` and `keepdims` arguments are passed directly to `numpy.linalg.norm` and are only usable if they are supported by the version of numpy in use.

  References

  [R93] | (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)
      -4.6566128774142013e-010
      >>> norm(b, -1)
      6
      >>> norm(a, 2)
      7.745966692414834
      >>> norm(b, 2)
      7.3484692283495345

      >>> norm(a, -2)
      0
      >>> norm(b, -2)
      1.8570331885190563e-016
      >>> norm(a, 3)
      5.8480354764257312
      >>> norm(a, -3)
      0