Math 2.0

Linear and multilinear algebra; matrix theory Definition

Companion matrix

Builds on Square matrixCharacteristic polynomial

Definition

Let FF be a field, let n≥1n \ge 1, and let p(t)=tn+cn−1tn−1+⋯+c1t+c0p(t) = t^n + c_{n-1} t^{n-1} + \cdots + c_1 t + c_0 be a monic polynomial over FF. The companion matrix of pp is the n×nn \times n matrix C(p)=(00⋯0−c010⋯0−c101⋯0−c2⋮⋱⋮00⋯1−cn−1).C(p) = \begin{pmatrix} 0 & 0 & \cdots & 0 & -c_0 \\ 1 & 0 & \cdots & 0 & -c_1 \\ 0 & 1 & \cdots & 0 & -c_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & 1 & -c_{n-1} \end{pmatrix}.

Remarks

The characteristic polynomial and the minimal polynomial of C(p)C(p) both equal pp. In the basis e1,…,ene_1, \dots, e_n, the map of C(p)C(p) sends eie_i to ei+1e_{i+1} for i<ni < n. Some authors use the transpose of this matrix as the companion matrix. Companion matrices are the blocks of the rational canonical form.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Companion_matrix, lead section
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