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The Companion Matrix

Suppose we have a nonconstant polynomial  
\[p( \lambda )=a_0 + a_1 \lambda + a_2 \lambda^2 +...+ a_n \lambda^n\]
.
The companion matrix associated with this polynomial is the square matrix with the diagonal below the leading diagonal consisting only of 1's, last column of the negative coefficients of  
\[\lambda^i , \: i=0,1,...,n-1\]
  and all other entries zero.
Example:  
\[ p( \lambda )=-2+3 \lambda +4 \lambda^2- \lambda^3\]
.
The companion matrix is  
\[ \left( \begin{array}{ccc} 0 & 0 & 2 \\ 1 & 0 & -3 \\ 0 & 1 & -4 \end{array} \right) \]

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