fix for eigenvectors function in eigen.mac

Hi all,

The eigenvectors() function in eigen.mac fails for a pretty simple 2x2
matrix, but there is a fix (see the bottom).

An example is shown below. In intro quantum mechanics a common problem
is to find the eigenvectors for an arbitrarily oriented Stern-Gerlach
apparatus. So one constructs:

Sn:[Sx,Sy,Sz].[sin(t)*cos(p),sin(t)*sin(p),cos(t)];

where Sx, etc. are the 2x2 spin matrices, and the angles t and p refer
to the usual spherical coordinates theta and phi. It is easy to
diagonalize this by hand.

The output from eigenvectors is

(%i1) display2d_unicode:false;
(%i2) load(qm);
(%o2)          /home/ehm/math/Maxima/share/ehm/qm-maxima/qm.mac
(%i3) Sn:[Sx,Sy,Sz].[sin(t)*cos(p),sin(t)*sin(p),cos(t)];
(%i4) [evals,evecs]:eigenvectors(Sn);
  
eigenvectors: the eigenvector(s) for the 1 th eigenvalue will be
missing. 
                        2       2         2       2         2
           hbar sqrt(sin (p) sin (t) + cos (p) sin (t) + cos (t))
(%o4) [[[- ------------------------------------------------------, 
                                     2
             2       2         2       2         2
hbar sqrt(sin (p) sin (t) + cos (p) sin (t) + cos (t))
------------------------------------------------------], [1, 1]], 
                          2
             %i p          2         2        2         2
[[], [[1, (%e     sqrt((sin (p) + cos (p)) sin (t) + cos (t))
                                       2         2
 + (- %i sin(p) - cos(p)) cos(t))/((sin (p) + cos (p)) sin(t))]]]]

And the first eigenvector is missing. Let's compare with my hand
written diagonalization code:

(%i5) [evals,evecs]:diagonalize(Sn)$

The output is a bit of a mess, so simplify it. (this is my fullsimp
available on github). This is the correct answer.

(%i6) fullsimp(%);
                            [        1        ]  [       1       ]
              hbar  hbar    [                 ]  [               ]
(%o6)     [[- ----, ----], [[     %i p     t  ], [   %i p     t  ]]]
               2     2      [ - %e     cot(-) ]  [ %e     tan(-) ]
                            [              2  ]  [            2  ]


The FIX:

To fix this in eigenvectors() replace algsys with linsolve.

Change the two lines

solution:algsys(equations,unknowns),
interm:map('rhs,solution[1]),

to

solution:linsolve(equations,unknowns),
interm:map('rhs,solution),

There are not many tests for eigenvectors in rtest_eigen.mac. Some more
tests should probably be added.

-Eric M