Companion Matrix

February 17, 2008 — Karthik

The companion matrix of the monic polynomial p(t)=c_0 + c_1 t + \dots + c_{n-1}t^{n-1} + t^n is the square matrix defined as:

C(p)=\begin{bmatrix} 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_{n-1} \\ \end{bmatrix}.

p(t) is also the characteristic polynomial for the matrix. The eigenvalues for the above matrix are the roots of p(t).

I found this to be quite an interesting way to find the roots of any equation, with a great deal of ease! The corresponding MATLAB function is compan.

\mathbf{F} = \textrm{compan}([1 -1 -1]) = \begin{pmatrix} 1 & 1\\ 1 & 0 \\\end{pmatrix}

\begin{pmatrix} F_{k+2} \\ F_{k+1} \\ \end{pmatrix} = \begin{pmatrix} 1 & 1\\ 1 & 0 \\\end{pmatrix} \begin{pmatrix} F_{k+1} \\ F_{k} \\ \end{pmatrix}

Here, F_k is the kth; Fibonacci number.

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