I am not a matrix theorist, or numerical linear algebra expert, but I have a problem and my proposed solution leads me to a question that I cannot answer.
I can give more details, but the gist is that I have a matrix $A$ which depends (smoothly) on a variable $x$. For a nice value of $x$, call it $x_0$, I can symbolically compute lots (but not all) about $A$ and its eigenstructure. In particular, I can find its leading eigenvalue $\lambda$, the corresponding eigenvector $v$, and I also know the derivative $K$ of the leading eigenvalue wrt $x$.
I want to know what happens when $x_0$ is perturbed to $x_0 + \epsilon$ where I don't care how small $\epsilon$ is. In particular, I want to know how the entries in the leading eigenvector change (relative to each other).
So what I want to do is to express everything as series expansions, truncated at the linear term, in $\epsilon$, so I will have
- $A$ becomes $A + \epsilon A'$ (where I know $A'$)
- $v$ becomes $v + \epsilon v'$ (where $v'$ is what I am trying to find)
- $\lambda$ becomes $\lambda + K \epsilon$ (because I know $K$)
Then I can solve the equation $$ (A + \epsilon A') (v + \epsilon v') = (\lambda + K \epsilon) (v + \epsilon v') $$ and determine $v'$, thus discovering how the entries in the leading eigenvector are perturbed as $x$ changes.
The only trouble is that I know that this general procedure is not always valid, because it makes various assumptions about the existence / convergence of these series expansions that are not always true for nasty matrices.
My matrix (at $x=x_0$) is probably not nasty, but I don't know how to tell for sure - I have tried leafing through Stewart & Sun, and even briefly attempted looking in Kato, but that made my head hurt.
But I somehow feel that an experienced matrix perturbation analyst would just be able to tell immediately whether this is safe or not, and hence my question:
Under what conditions on the matrix is it legitimate to analyse perturbations via the "expand in series and truncate" method outlined above?
Here are some additional facts about my matrix that may or may not be of importance.
Things I feel are probably good
- at $x_0$ it has real eigenvalues
- at $x_0$ the leading eigenvalue is simple
- at $x_0$ the matrix is diagonalisable
Things that may not be so good
- at $x_0$ the matrix has a 6-dimensional nullspace (but all other eigenvalues are simple)
- the matrix is not symmetric, nor is it non-negative
Things that are probably good, but that I can only see empirically (i.e. by computer)
- the matrix has distinct real eigenvalues throughout some interval of the form $(x_0, x_0+\epsilon)$
- working to high precision with Mathematica, everything that I have said above about how terms vary can be validated for specific small values of $\epsilon$