Hi ArxIv,
Expectations might be too high for symbolic solutions of eigenvectors.
Consider a simpler, but very closely related problem:
a11 = 5;a22 = 10;
syms a12
A = [a11,a12;a12,a22];
[V,D] = eig(A);
diag(D)
disp(char(ans))
V = simplify(V)
disp(char(V))
I believe that the eigenvalues will yield the correct result for any value of a12, as will the eigenvectors except for the problematic case if we attempt to evaluate them at a12 = 0. The question is if there exists a general expression for V that works for all possible values of a12. If not, then the ideal solution would be for eig to return a piecewise expression of some sort for V taking into account all possible special cases. Other functions, like int also typically do not identify and return piecewise solutions for special cases. I'd imagine it may be difficult to find all special cases for more complicated problems. So I guess eig does the best it can.
Sometimes, we can get the solution by taking the limit as a12 ->0, which works in this case for V(:,2)
limit(V(:,2),a12,0),disp(char(ans))
But that "trick" doesn't work as well for V(:,1)
limit(V(:,1),a12,0),disp(char(ans))
though it kind of does if we realize that V(1,1) is blowing up to infinity, and so the normalized eigenvector is, in fact, approaching [1;0].
syms a11 a22 real
sort([a11,a22],'descend'),disp(char(ans))
Is there any mathematical reason to expect a22 >= a11? What happens if we sub values for a11 and a22 into that result when the latter is smaller than the former?
assumeAlso(a11 > a22);
sort([a11,a22],'descend'),disp(char(ans))
