"1^(x=defective square_matrix)" does not evaluate to I(dim(x))

[Ben Abbott]

On Monday, January 14, 2008, at 11:39AM, "Rolf Fabian" <Rolf.Fabian at gmx.de> wrote:
>
>It can be shown, that
>1^(x = square matrix of size n) should always evaluate to
>eye(n), no matter of the contents of exponent matrix x
>as long as no NaN elements occur in x.
>
>octave-3.0.0.exe:> n=1; 1^(x = randn(n)+rand(n)*i)   # OK
>ans =  1
>octave-3.0.0.exe:> n=2; 1^(x = randn(n)+rand(n)*i)   # OK
>ans =
>   1.00000 - 0.00000i   0.00000 + 0.00000i
>   0.00000 - 0.00000i   1.00000 + 0.00000i
>
>octave-3.0.0.exe:> n=3; 1^(x = randn(n)+rand(n)*i)   # OK
>ans =
>   1.00000 + 0.00000i   0.00000 + 0.00000i   0.00000 + 0.00000i
>   0.00000 + 0.00000i   1.00000 + 0.00000i   0.00000 + 0.00000i
>   0.00000 + 0.00000i   0.00000 + 0.00000i   1.00000 + 0.00000i
>
>octave-3.0.0.exe:> x=[0 1 0 0; 1 0 2 3; 0 0 0 1; 0 0 1 0]
>x =
>   0   1   0   0
>   1   0   2   3
>   0   0   0   1
>   0   0   1   0
>
>==============================
>octave-3.0.0.exe:> 1^x                              # FAIL
>ans =
>   1.00000   0.00000   1.37500   1.62500
>   0.00000   1.00000   2.00000   1.00000
>   0.00000   0.00000   1.00000   0.00000
>   0.00000   0.00000   0.00000   1.00000
>==============================
>
>This result differs significantly from expected result,
>namely 'eye(4)'.
>Hence the evaluation by Octave is definitely incorrect.
>BTW What does MatLab report here ?

Matlab 2007b on my Mac PPC gives,

>> x=[0 1 0 0; 1 0 2 3; 0 0 0 1; 0 0 1 0];
>> 1^x
Warning: Matrix is close to singular or badly scaled.
         Results may be inaccurate. RCOND = 2.220446e-17.

ans =

   1.000000000000000                   0   0.366116523516816   0.366116523516816
                   0   1.000000000000000                   0                   0
   0.000000000000000  -0.000000000000000   1.000000000000000  -0.000000000000000
                   0                   0                   0   1.000000000000000

>> 

Ben
>
>What is special with the above counter example x ?
>At a first glance nothing, except that it has only
>zero entries on the main diagonal and thus it is
>traceless. It's determinant is det(x)=1, thus it
>is regular (non-singular).
>
>But x is severly defective which can be checked
>by
>octave-3.0.0.exe:> [v,d] = eig(x)
>v =
>   0.70711  -0.70711  -0.70711   0.70711
>   0.70711   0.70711  -0.70711  -0.70711
>   0.00000  -0.00000   0.00000  -0.00000
>   0.00000  -0.00000   0.00000   0.00000
>
>d =
>   1   0   0   0
>   0  -1   0   0
>   0   0   1   0
>   0   0   0  -1
>
>octave-3.0.0.exe:> cond(v)
>ans = 2.2960e+016
>octave-3.0.0.exe:> rank(v)
>ans =  2
>
>There is a rank deficiency of 2 for the
>matrix of eigenvectors v . 
>Thus v is not invertible and the
>obviously underlying similarity
>transformation fails.
>
>=========================
>Proposed receipe of a fix,
> c : real or complex scalar :
> c ^ (x = any square matrix)
> = log ( c ) .* eye ( dimension (x) )
>no matter about particular
>content of matrix x ( as long
>as input does not contain 'NaN',
>but  -/+ Inf elements of x 
>should be allowed / accepted.
>=========================
>
>Rolf Fabian
>
>< r dot fabian at jacobs-university dot de>
>
>
>-----
>Rolf Fabian
><r dot fabian at jacobs-university dot de>
>
>-- 
>View this message in context: http://www.nabble.com/%221%5E%28x%3Ddefective-square_matrix%29%22-does-not-evaluate-to-I%28dim%28x%29%29-tp14804976p14804976.html
>Sent from the Octave - Bugs mailing list archive at Nabble.com.
>
>_______________________________________________
>Bug-octave mailing list
>Bug-octave at octave.org
>https://www.cae.wisc.edu/mailman/listinfo/bug-octave
>
>