I have a doubt in linear algebra basically about polynomials.
If a sequence of real matrices $A_n$ converges to a matrix $A$, does it imply that in $\mathbb{C}^n$, the spectrum vectors $\sigma_n$ (by which I mean a vector with coordinates are eigen values of $A_n$) converges to $\sigma$ (The spectrum vector of $A$)?
Thanks in Advance.
There's no natural ordering on the eigenvalues. The coefficients of the characteristic polynomials $P_i$ certainly converge to the coefficients of $P$, but does that mean that the "roots converge"?
It's probably better to put it this way: the coefficients of a polynomial are related to the elementary symmetric functions in its roots. So yes, the elementary symmetric functions for the eigenvalues do converge to the correct thing. You should think of these symmetric functions as the "coordinates" for the space of eigenvalues—the eigenvalues themselves are not coordinates, because nothing happens when we switch them around.