For a general parameter (t) dependent m x m matrix, M(t), can one always diagonalize it by evaluating
D (t)=U(t)^{-1}M(t)U(t),
where D(t) is a diagonal matrix. Under what conditions can I numerically evaluate U(t) such that the parametric dependence is retained either exactly or to a very good approximation.
I would also like to know some standard algorithms or approximations. In case I am conceptually not right about thinking this way, please point out the error.
Matrices are diagonalizable if they are normal, i.e. (A^*)A = A (A^*). I don't know much about approximating parametric eigenvectors, but if it is useful at all, the "Successive Constraint Method" is an efficient algorithm approximating the smallest eigenvalues of a parametric matrix.
Diagonalization, as you see it (by the presented recipe), is known as Jacobi method. In its practical realization you don't see the matrix U explicitly. This method has such an advantage over other methods that it delivers different eigenvectors for identical (degenerate) eigenvalues. But as you ask for numerical calculations, I don't see why you can't construct matrices M(t) for various values of parameter t and see the effect in corresponding eigenvectors? I'm probably missing something ...
@ Marek. I have constructed matrices for various t values. And I can fit its behavior with some polynomial in t. I am looking for some other method which is not curve-fitting so as to compare the regimes where the fit doesn't work well.
I suggest you to read the paper of H.J.Korsch and M.Gluck in EJP. I hope this paper will help you. However you can calculate the wafe function using perturbation theory analytically .In this case see the work of Morse and Feshbach.
B.Rath
@Biswanath Rath: The paper provides a recipe to calculate the eigenvalues. I found Korsch and Gluck's codes pretty simple and effective for some few well known problems. However, once the eigenvalues are known, one has to fit it with a some polynomial to get back the parameter dependence in order to calculate the eigenvectors. Else, I will just have snapshots of the eigenvectors at different parameter values. This is what Marek has suggested. I want a way to bypass this fitting procedure and express the eigenvalues as well approximated functions of the parameter and then derive the eigenvectors. Thank you.
It is quite important to know whether your matrices are symmetric (or hermitian if complex). If so, you can always diagonalise them. Otherwise, there can arise the phenomenon of Jordan blocks, in which an eigenvalue has multiplicity 2, say, and only one eigenvector. If that happens, or even if you are close to such a point, the sensitivity of the eigenvectors can be huge.
Contact prof s.n biswas (retd ) or see the j math physics ,if I remember he has
Calculated wave function using hill det method.however best way to use perturbation theory.
The two most useful criteria (sufficient conditions) for diagonalizability are (1) that A commutes with its conjugate transpose (normal -- already mentioned) and (2) that the eigenvalues of A are simple (roots of the characteristic polynomial with multiplicity 1). Indeed, the eigenvector for a simple eigenvalue (and the eigenvalue itself) will depend differentiably on the matrix, so presumably on your parameter, and the computation should work nicely as a perturbation problem. [This extends to suitable linear operators on infinite-dimensional spaces.] I would recommend Kato's book for the theory, but have no specific recommendation for the computation, which depends strongly on the available known structure of the parametrized matrix.
A matrix A being normal means AA^*=A^'A or, equivalently, that A can be diagonalized via unitary transformations. The word unitary is key here. There are nonnormal A which can still be diagonalized by similarity! For instance, take every triangular matrix with distinct diagonal entries. In general, A is diagonalizable (by similarity) if the algebraic and geometric multiplicities coincide for all eigenvalues of A. This includes multiple eigenvalues having only Jordan blocks of size 1.
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