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

[Rolf Fabian]

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 ?

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>

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