This is my first time posting a question here, so if there's anything about the way I'm posting that I can improve upon, please let me know.
I have to give a presentation on root bounds for polynomials that can be derived from Gershgorins circle theorem and I was given the following excerpt: https://www.jstor.org/stable/2313703?seq=1.
In this paper, there are a bunch of proofs for different root bounds using Gershgorins theorem and I am struggling to comprehend some of them.
The basic idea for most of these proofs is to form the companion matrix $C(f)$ for a polynomial $f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0 $, defining a diagonal matrix $P = diag(b_1,\cdots,b_n)$ and then forming the matrix $P^{-1}CP$, whose eigenvalues are the roots of $f$.
My question is regarding the proof of Cauchys theorem, which states that the moduli of the roots of a polynomial $f$ are less or equal to the positive root of $g(x) = |a_n| x^n - |a_{n-1}|x^{n-1} - \cdots - |a_0|$. First, we set the entries $b_i$ of the diagonal matrix $P$ to be $b_i = \rho^{n-i}$, which gives us $ P^{-1}CP = \begin{pmatrix} 0 & 0 & \cdots & 0 & -a_0 / (a_n \rho^{n-1}) \\ \rho & 0 & \cdots & 0 & -a_1 / (a_n \rho^{n-2}) \\ \vdots & \rho & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & -a_{n-2} / (a_n \rho) \\ 0 & 0 & \cdots & \rho & -a_{n-1} / a_n\\ \end{pmatrix} $ Now, according to the paper, we're supposed to apply Gershgorins theorem to the columns of $P^{-1}CP$ and chose a $\rho$ thats equal to the sum of the moduli of the elements in the last column of the matrix. How does this conclude the proof? And how can I chose $\rho$ to be equal to the sum of the moduli of the last column, when $\rho$ is actually a factor in all those terms?
One more thing: Apparently, if only the moduli of the coefficients of a given polynomial are known, there cant be a smaller bound than this one by Cauchy (correct me if Im wrong), but how would I go about proving this? I dont need a mathematically sound proof here, just a rough direction of why this holds.
Thank you!!
