Consider a symmetric matrix $A$ with non-negative integer coefficients. It appears that the geometric series $\sum_{i \geq 0}A^i$ will converge to a matrix $B$ if the spectral radius (the largest eigenvalue in absolute value) is less than 1.
I would like to extend this result to a case where the matrix doesn't have integer coefficients but generating functions (with real coefficients) instead and to do so, I'd be interested in having a closer look at the proof of the standard case.
The questions I'm trying to figure out are:
1) is there a standard proof in a general case where we're dealing with a ring with a norm, than the geometric series has 1 as radius of convergence? Or what's the most obvious proof in the specific case of symmetric non-negative matrices?
2) what is the good equivalent of the case of matrices with integer coefficients that aren't necessarily symmetric. I'm quite interested in keeping the link to the eigenvalues and not in other matrix norms.
3) when dealing with matrices of generating functions (on one variable for a start), can I think of the eigenvalues as generating functions as well? Would that be the good way to think of this result on geometric series?
I'll keep posting as I work on all this (but I needed a place to write things down and possible get some help or hints).
Answer
In any Banach algebra with identity over $\mathbb C$, the spectral radius of $x$ is $\rho(x) = \lim_{n \to \infty} \|x^n \|^{1/n}$. If $|t| \rho(x) < 1$, the geometric series $\sum_{n=0}^\infty t^n x^n$ converges to $(1 - t x)^{-1}$. This is standard material that you can find in any text that discusses Banach algebras.
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