# Solving for eigenvectors with eigenvalues

I'm not good with math. I can say I understand the big picture, but I rely on Matlab for doing everything for me.

I have a specific problem:

$$US = \left(\theta F + (1-\theta)F XX^\mathrm{T} F\right)S,$$

where $F$ is an $N \times N$ matrix of ones, $X$ is unknown of length $N$, $X^\mathrm{T}$ is the transpose of $X$, $U$ is the eigenvalues of $S$ (which is unknown and of length $N$), $\theta$ is a scalar value and is unknown too.

This matrix is of rank $1$, $N-1$ of the eigenvalues are $0$ and the last is $1$.

The big problem is to find $S$, which I have no idea how to do so, but the paper that mentions the problem also mentions that a step in finding $S$ is to find the eigenvectors of $S$ (If I got that right).

The expected solution of $S$ is a set solutions not just on. $X$ is known to be of the same set, but $X$ and $S$ are not the same every time (same set of solutions but different solution).

So I tried that with Matlab as follows:

I defined syms vectors ($S$ and $X$, of length $8 \times 1$) and defined $\theta$ as syms too. I used the eig() function to find the eigenvalues and eigenvectors,it returned eigenvectors that seem right, $7$ of them are zeros and the last one is an expressions of all syms variables (which I guess equals $1$). The eigenvectors however are strange, they are independent of all syms variable, I tried to change some of these variables with random values set by me, but I always get the same eigenvalues.

It is obvious that changing $\theta$ in the first part of the equation ($\theta F$) would change that part's eigenvectors for $S$, which means changing the other part should change its eigenvectors for $S$ in a reversed manner, but the second part does not, which drove me to think that eig() is not the right function to deal with a problem like this.

I have two questions now:

1. How do I get the right eigenvectors (they should be dependent on the syms variables, some of them at least?
2. Or how do I solve for $S$ with all these unknowns?
• Try $S=0$? Your question is just a little vague. – copper.hat Dec 17 '13 at 23:18
• what do you mean try S as zero? S is the solution I'm looking for. I'm also sorry If my question is vague, this is as much as I know about the problem, I tried to explain it the best way I could. – ProWi Dec 17 '13 at 23:21
• I mean, substituting $S=0$ will solve the problem as far as I can tell? – copper.hat Dec 17 '13 at 23:25
• I still don't get that,how can I substitute S when I'm trying to find it? – ProWi Dec 18 '13 at 14:38
• The value $S=0$ will solve the problem, which presumably is not interesting to you. So, you need some conditions that require $S$ to be non-zero. – copper.hat Dec 18 '13 at 15:53