Why are the components of the two eigenvectors not continuous and s... (2024)

Note that eigenvectors are not generally continuous functions of the matrix entries, as shown in this old example from J.W. Givens:

The matrix H(k) is continuous as a function of k, even at k=0, but its matrix of eigenvectors U is not:

I believe this can only occur in a neighborhood of H where eigenvalues have multiplicity greater than 1. Based on your plots, it is not clear whether this happens for your data. In general, though, it should be kept in mind that the eigenvalues of a matrix , but not necessarily the eigenvectors, are continuous in the matrix elements.

You can think of the eigenvalues (and assiciated eigenvectors) as coming from the roots of a polynomial. In fact, they do exactly that. They are derived from a characteristic polynomial. Now, suppose you have a polynomial where the coefficients are a function themselves of some parameter. The roots will vary, as a function of that parameter. The problem is, those roots (as well as the function roots) do not understand the parameter they depend upon. That are just numbers. The function roots does not understand that parameter. It gives not a tinker's damn why it is called, only that you pass it a vector of coefficients, and it returns a valid result.

For example, suppose we consider a simple polynomial in x. I'll just pick something at random. Again, while this is a polynomial, I could as easily be creating a matrix, and computing the eigenvalues. There is a one to one correspondence here. We can always create a matrix that has the same eigenvalues as any polynomial (this is called the companion matrix, created by the matlab function compan), and we can always take any matrix and generate the characteristic polynomial. So what I do here is entirely valid as a comparison. I'll show it in symbolic form first.

I've been very arbitrary in that polynomial, and honestly, I have not a clue what it looks like, as a function of t, and where the roots might lie, nor how they might vary.

Hmm. Coeffs decided to put them in the opposite order of what roots would want., and for some stupid reason, flip does not fix the issue. I care not, and I won't bother fixing it. ;-)

For any value of t, we can comput the roots, and follow them, as they are returned from roots. Again, roots does not care about the order of those roots, or how that order is related to the parameter t. You NEED to understand that. It does not think of the roots as a function of t. Only you think that way. and roots gives not a hoot about what you understand.

Now, plot the roots, in sequence, exactly as they were returned.

Do you see the roots bounce around in order? If, instead, I resequence those roots, they would surely follow nice paths in the complex plane. Using my eigenshuffle code (Found on the file exchange, it can improve on that mess, though even it is not perfect.)

Here you can see the roots do indeed follow well-behaved paths around the complex plane, though eigenshuffle was not perfect in its sequencing choice.

Again, you need to understand that eig does not know how the eigenvectors and eigenvalues should vary as the elements of a matrix vary.