I have two related questions. My question is hypothetical, i.e. not from an actual physical problem. If I give you a matrix and tell you that it has a repeated eigenvalue, can you say anything about whether the eigenvectors of that repeated eigenvalue are linearly independent or not? Is there a general procedure to check? I don't know the elements of the matrix itself so I can't work out the eigenvectors in the usual way.
On that note, can someone provide me an example of a matrix with a repeated eigenvalue but where the eigenvectors are not linearly independent?
Assume the field is complex numbers.
Best Answer
Once we have one eigenvector $v$ corresponding to the eigenvalue $\lambda$, any non-zero scalar multiple of $v$ is also an eigenvector corresponding to $\lambda$. So rather than talking about the number of eigenvectors, we talk about the number of linearly independent eigenvectors ( for example corresponding to a particular eigenvalue), equivalently the dimensions of the eigenspaces (corresponding to a particular eigenvalue).
One useful thing is that for an eigenvalue, its geometric multiplicity cannot exceed its algebraic multiplicity. In general, one is often forced to solve for the eigenspace. Depending on the information you have, you may be able to make some deductions without solving for the eigenspace/s.
Again in regard to your last question, you need to be clear on what you are asking.
For example $A = \left(\begin{matrix} 1 & 1\\0 & 1\end{matrix}\right)$ has one repeated eigenvalue (multiplicity two) $\lambda =1$. Solving for the corresponding eigenspace, we get $v = t \left(\begin{matrix} 1\\0\end{matrix}\right)$, $t\in\mathbb{R}$. So we can write down only one eigenvector, if we are asked to give the set of linearly independent eigenvectors (any nonzero scaler multiple of $\left(\begin{matrix} 1\\0\end{matrix}\right)$ will do).
On the other hand, the $2\times 2$ identity matrix again has only one eigenvalue (multiplicity two) but its corresponding eigenspace has dimension $2$ (note any nonzero vector is an eigenvector). So we have two linearly independent eigenvectors.