Re: fishy rank

[Etienne Grossmann]

  Hello,

From: Daniel Heiserer <address@hidden>
#  I have situations where 

#  :rank(A)>rank([A,B]);

#  also 
#  :rank(A)>=rank(A^n)
#  if n is large enough.
[snip]
# Can this be seen as a numerical problem in
# terms of conditioning?

  Could be because

  (singular values of A^n) == (sing. vals. of A).^2 

  so that the ratio of small/big sing. val. will tend to zero. If A
has all sing. vals. equal to zero or one, there is no problem (see
below).

  Cheers,

  Etienne

======================================================================
octave:23> A = randn (5,3); [U,D,V] = svd (A,"cheap"); A = U*U';
octave:24> rA = rank (A)
rA = 3
octave:25> for i = 1:100:1000, if rank (A^i) != rA, [i, rank (A^i)], end; end
octave:26> A = randn (5,3); A = A*A';
octave:27> rA = rank (A)
rA = 3
octave:28> for i = 1:10:100, if rank (A^i) != rA, [i, rank (A^i)], end; end
ans =

  21  2

ans =

  31  1

ans =

  41  1

ans =

  51  1

ans =

  61  1

ans =

  71  1

ans =

  81  1

ans =

  91  1



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