Colleague Matrix Can someone explain to me the concept of a Colleague Matrix. I tried to find some information online and I haven't been able to find anything. 
Example..
Given the function
$$f (x) = x\bigg(x − {1\over4}\bigg)\bigg(x − {1\over2}\bigg)$$
show that its colleague matrix is given by
$C = \begin{bmatrix}0 & 1 & 0\\{1\over2} & 0 & {1\over2}\\{3\over4} & -{5\over4} & {3\over4}\end{bmatrix}$
 A: I have mixed feelings when I found that this unanswered thread is among the first few results returned by Google when searching for "colleague matrix". Here's my attempt at making this search result more useful than it currently is.
The term "colleague matrix" was coined by I. J. Good in this paper; it is one of a family of so-called "comrade matrices" that are completely analogous to the classical Frobenius companion matrix.
Briefly put, if a polynomial is expressed entirely in terms of an orthogonal polynomial basis $p_k(x)$ (e.g. $w(x)=c_0 p_0(x) + c_1 p_1(x) + \cdots + c_n p_n(x)$), the comrade matrix is a rank-1 correction to the tridiagonal matrix formed by the recurrence coefficients of the $p_k(x)$, where the rank-1 correction is formed from the coefficients $c_k$. If the orthogonal polynomial basis chosen is the Chebyshev polynomial of the first kind, $T_k(x)$, the matrix is termed a "colleague matrix".
They serve the same purpose as the Frobenius matrix; that is, the characteristic polynomial of the comrade matrix corresponding to $w(x)$ is in fact $w(x)$. This is useful in the rootfinding case, because there are a number of reasons why converting a polynomial expressed as an orthogonal series to the monomial basis can be A VERY BAD IDEA, and one can now use any of a number of matrix eigenvalue methods (e.g. the Francis QR algorithm) to find the roots of $w(x)$, through its comrade matrix.
