Matrix Definition of Cyclic Subspace

Theorem

• Let $V$ as a vector space
• Let $\alpha \in V$
• With $T\in \mathcal{L}(V)$
• Then $p(x)=x^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$ be the $T$-annihilator of $\alpha$
• With $\beta =\{\alpha ,T\alpha ,\dots ,T^{n-1}\alpha \}$ as a basis for $Z({\alpha }_{i}T)$ be the linear operator defined by restriction of $T$ onto the subspace $Z({\alpha }_{i}T)$ then,

[T_{Z}]{\beta} = \left[\begin{array}{cc} 0 & 0 & 0 & \dots & 0 & - \alpha{0}\ 1 & 0 & 0 & \dots & 0 & -\alpha_{1}\ 0 & 1 & 0 & \dots & 0 & - \alpha_{2}\ \vdots & \vdots & \ddots & \ddots & 0 & -\alpha_{z}\ 0 & 0 & 0 & \dots & 1 & -\alpha_{z-1}\ \end{array}\right]

# Example Find companion matrix for $x^{3} -5x + 8x - 4$ 1. We write the first columns out

\left[\begin{array}{ccc} 0 & 0 & ?\ 1 & 0 & ?\ 0 & 1 & ?\ \end{array}\right]

$$2.Thelastcolumnisthenegativeversionsofourcoefficients$$

\left[\begin{array}{ccc} 0 & 0 & 4\ 1 & 0 & -8\ 0 & 1 & 5\ \end{array}\right]