I am studying the theory of random matrices lately, but there is a basic issue troubling my life. I hope someone here explain me this, thank you.
A random matrix is defined as a matrix whose entries are random variables. That is ok, but then they start to talk about the eigenvalues of random matrices like it's a normal thing, no explanation at all about this. The problem is: the entries are random variables, not numbers, and this changes everything.
If $X$ is a random matrix and $\lambda$ is an eigenvalue of $X$, then there is an eigenvector such that $$X\cdot v = \lambda v.$$
When doing the multiplication, I am multiplying random variables by the coordinates of $v$, are this coordinates numbers? Or Random variables too? And $\lambda$, it is a number or a random number (in this case $\lambda$ would be a random variable too I think) ?
PS: I would be very grateful if someone shows some explicit example, to clarify this ideas.
Thanks.
Word after years: Visiting this topic again, after years studying, I realize better what was the "trigger" which caused all this problem.
First, they use the term eigenvalues, not random eigenvalues. So I was taking the literal definition and thought they were numbers, not random variables. For someone working in the field it should be obvious they were random variables since they came from random matrices. I was very novice in this area, in fact, I was studying the very first definition of random variables a few days before.
Second, at the time I saw "$\lambda_1 \leq \ldots \leq \lambda_N$" in the book, I really wasn't used to consider ordering over functions, to write things like $f \leq g$, when $f,g$ are functions. Ordering is something for numbers, was my point of view that time. Again, this was due to my lack of experience.
Best Answer
If $X$ is a random matrix, the eigenvectors and eigenvalues of $X$ are random as well. Thus the vector $v$ and the real number $\lambda$ in your example are random.
Edit: At the risk of belaboring the obvious, in this context, the random matrix $X$ is a function $X:\Omega\to\mathcal M_n(\mathbb R)$ and an eigenvector and an eigenvalue of $X$ are functions $v:\Omega\to\mathbb R^n\setminus\{0\}$ and $\lambda:\Omega\to\mathbb C$ such that, for every $\omega$ in $\Omega$, $X(\omega)v(\omega)=\lambda(\omega)v(\omega)$. For example, the first coordinate of the vector $X(\omega)v(\omega)$ is $\lambda(\omega)v_1(\omega)$.