How to find the companion form of a matrix from its eigenvalues?
In the literature, there are a lot of explanations about finding the jordan canonical form of a matrix. However, there is almost nothing about finding a matrix from its eigenvalues, especially multiple eigenvalues. I need to obtain the transpose companion form (Bush form) of a matrix whose eigen values are given. For example how can I find the companion form of a matrix A whose eigen values are -1,-1, and -1 (i.e. three repeated eigenvalues.)
We know that S^-1*A*S=J. Since we know the eigen values, we can write J. And A=S*J*S^-1. There are some formulas to find S when the eigen values are distinct (e.g. S=[1 1 1; e1 e2 e3; e1^2 e2^2 e3^2] for a 3x3 matrix). However, when the eigen values are multiple this formulas fail. For such a case, how can we obtain S?
To find a general formulation for our case I tried to make some calculations. I found a formula for two repeated and one distinct eigen values case. i.e. When eigen values are e1 e1 e3, S=[1 0 1; e1 1 e3; e1^2 2*e1 e3^2]. However, for triple eigen values case (e1 e1 e1) I couldn't find any solution.
If there is some having an idea, please share it :)