JULIA matrix norm is different MATLAB

General Usage
6

Actually, the previous posts have it backwards, as far as the names of the norms are concerned. For matrices, it’s opnorm that gives you the induced matrix 2-norm, \|A\|_2 = \sup_{x\neq0} \|Ax\|_2/\|x\|_2 = max singular value, whereas norm gives the root-sum-squares Frobenius norm, \|A\|_F = \sqrt{\sum_{i,j} A^2_{ij}}.

E.g.

julia> A = [1 1; 2 0]
2×2 Matrix{Int64}:
 1  1
 2  0

julia> opnorm(A)
2.2882456112707374

julia> svdvals(A)
2-element Vector{Float64}:
 2.2882456112707374
 0.8740320488976421

julia> norm(A)
2.449489742783178

julia> sqrt(1^2 + 1^2 + 2^2 + 0^2)
2.449489742783178

That’s what documentation says as well

help?> opnorm
search: opnorm

  opnorm(A::AbstractMatrix, p::Real=2)

  Compute the operator norm (or matrix norm) induced by the vector p-norm, where valid values of p are 1,
  2, or Inf. (Note that for sparse matrices, p=2 is currently not implemented.) Use norm to compute the
  Frobenius norm.

and

help?> norm
search: norm normpath normalize normalize! opnorm issubnormal UniformScaling ColumnNorm set_zero_subnormals

  norm(A, p::Real=2)

  For any iterable container A (including arrays of any dimension) of numbers (or any element type for
  which norm is defined), compute the p-norm (defaulting to p=2) as if A were a vector of the
  corresponding length.

Note the “as if A were a vector of the corresponding length.” I.e. unpack A into a vector and compute the 2-norm of that vector. That is totally different from the induced matrix 2-norm.

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