I am still using MC v.11.x.
From a matrix I needed to extract the eigenvalues and eigenvectors, and re-order them (starting from the eigenvalues) in a decreasing manner. Accordingly I needed to change column order to the eigenvectors, so they correspond to the new order of the eigenvalues. I used the function eigenvec(M,z) for this. Yet without success.
The Figure I hope explains the problem, in its essence (there is no re-ordering here). It shows that the connected eigenvector to its eigenvalue (captured with "eigenvec" function), is not the same of the one that appears in the function "eigenvecs(M)".
I understand (or to say better, I realized, see the Figure) that the same eigenvalue may have multiple eigenvectors. But now:
1. How can I extract the same eigenvector that appers in the "eigenvecs" function, by recalling its own eigenvalue?
2. And more in general, is there a way to see the multiplicity of eigenvectors connected to a given eigenvalue?
Thanks.
Probably a good answer to my problem and doubts comes from an old post by @:
Eigenvectors, Eigenvals, Matrix
I quote the relevant passages in italics, with in bold something I mentioned above or that seems helpful for my task:
There is a function (eigenvec) that takes a matrix and an eigenvalue and returns an eigenvector associated with that eigenvalue. But unless the eigenvalues are well separated it is not reliable, and cannot return more than one eigenvector for an eigenvalue that is repeated.
Eigenvals and eigenvecs are defined to return the eigenvalues and eigenvectors in the same order (this may not work with the symbolic processor, which tends to have its own ideas of how to do things). If you want to sort the eigenvalues you should stack them onto the eigenvector matrix and then sort the matrix on the resulting eigenvalues row. That keeps them together.
Therefore the solution could be the following:
1. use eigenvecs and eigenvals
2. stack the Transpose eigenvalue vector onto the eigenvector matrix
3. use now csort rsort in the augmented matrix of point 2. (in the example above would be a 4x3 matrix), taken the column last row of eigenvalues as reference
4. in my case I need from high to low values, so revert the order: I use a diagonal matrix
5.extract the new ordered eivenvectors (and eigenvalues) with submatrix
Still somewhat puzzled by the eigenvec function: the eigenvector result, that it gives in the example, should be a linear combination of the ones given by eigenvecs function. But that it is a bit confusing since one expects linear independent eigenvectors.
That works 
. The only thing I can suggest changing is to transpose Q_Evec and augment it with EVal. Then use csort, followed by the reverse function to reverse the rows. I think it's a little neater to use the reverse function rather than multiplying by the diagonal matrix.
Still somewhat puzzled by the eigenvec function: the eigenvector result, that it gives in the example, should be a linear combination of the ones given by eigenvecs function. But that it is a bit confusing since one expects linear independent eigenvectors.
I'm not sure what you mean. Given three linearly independent vectors, I can generate an infinite number of other sets of equivalent linearly independent vectors by creating linear combinations of the ones I have.
