# 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

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

• 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
• 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
• This seems to be a better version: math.washington.edu/Seminars/Milliman/TrefethenDay2_handout.pdf – Martin Sleziak Nov 13 '12 at 11:33

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.