A note on companion matrices (Q1410726)

The companion matrix \(A\) of the monic polynomial \[ p(x)=x^n+a_1x^{n-1}+ \cdots +a_{n-1}x+a_n \] has only entries \(0, 1\), and \(a_i\) for \(i=1,\ldots,n\). The author gives several tridiagonal matrices that are similar to \(A\), also with entries \(0, 1\), and \(a_i\) only. He shows further that each of these tridiagonal matrices is a product of several simpler matrices. The new versions of the companion matrix can be used in estimates of the roots of the polynomial and perhaps their computation.