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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

$C = \begin{bmatrix}0 & 1 & 0\\{1\over2} & 0 & {1\over2}\\{3\over4} & -{5\over4} & {3\over4}\end{bmatrix}$

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As a general rule, if there's neither a Wikipedia article nor a MathWorld entry for a term you use and a Google search doesn't lead to a well-known site that has a definition, it makes sense to include a definition of the term in your question. –  joriki Nov 13 '12 at 11:10
    
That's the thing, I don't know what the definition is. This question is in one of my problem sets but I can't find anything on the subject. –  StealzHelium Nov 13 '12 at 11:20
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From this text: Theorem 18.1. Polynomial roots and colleague matrix eigenvalues. The roots of the polynomial $$ p(x) = \sum_{k=0}^n a_k T_k(x),\quad a_n \ne 0 $$ are the eigenvalues of the matrix $$C=\begin{pmatrix} 0&1\\ {1\over 2}&0&{1\over 2}\\ &{1\over 2}&0&{1\over 2}\\ &&\ddots&\ddots&\ddots\\ &&&&&{1\over 2}\\ &&&&{1\over 2}&0 \end{pmatrix} - {1\over 2 a_n} \begin{pmatrix} 0 & 0 & 0 & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & \dots & 0 \\ a_0 & a_1 & a_2 & \dots & a_{n-1}\end{pmatrix} . $$ –  Martin Sleziak Nov 13 '12 at 11:26
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This seems to be a better version: math.washington.edu/Seminars/Milliman/TrefethenDay2_handout.pdf –  Martin Sleziak Nov 13 '12 at 11:33

1 Answer 1

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.

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