#### Previous topic

scipy.linalg.det

#### Next topic

scipy.linalg.lstsq

# scipy.linalg.norm

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

  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. ord : {non-zero int, inf, -inf, ‘fro’}, optional Order of the norm (see table under `Notes`). inf means numpy’s inf object.
  Returns: | norm : float Norm of the matrix or vector.

  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 [R86]:

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

  References

  [R86] | (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
      >>> 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)
      nan
      >>> norm(b, -2)
      1.8570331885190563e-016
      >>> norm(a, 3)
      5.8480354764257312
      >>> norm(a, -3)
      nan