Section 31.2 Terminology and notation
- nilpotent matrix
a square matrix powers of which are eventually the zero matrix
- degree of nilpotency
the lowest positive exponent that takes a nilpotent matrix to zero; i.e. the positive number \(k\) for nilpotent \(N\) so that \(N^k = \zerovec\) but \(N^{k-1} \neq \zerovec\)
- companion matrix (of a monic polynomial)
-
the \(n \times n\) matrix
\begin{equation*} \begin{bmatrix} 0 \amp 0 \amp 0 \amp \cdots \amp 0 \amp -a_0 \\ 1 \amp 0 \amp 0 \amp \cdots \amp 0 \amp -a_1 \\ 0 \amp 1 \amp 0 \amp \cdots \amp 0 \amp -a_2 \\ \vdots \amp \vdots \amp \vdots \amp \amp \vdots \amp \vdots \\ 0 \amp 0 \amp 0 \amp \cdots \amp 1 \amp -a_{n-1} \end{bmatrix}\text{,} \end{equation*}where \(n\) is the degree and \(a_0,a_1,\dotsc,a_{n-1}\) the coefficients of the polynomial (with coefficient \(a_j\) corresponding to power \(x^j\))