Examples of T-conductor

examples-counterexampleslinear algebralinear-transformations

I am struggling with understanding the concept of a T-conductor in linear algebra. I do know the definition, but some examples would be helpful.

Definition. Let $W$ be an invariant subspace for $T$ and let $\alpha$ be a vector in $V$. The $T$-conductor of $\alpha$ into $W$ is the set $S_T(\alpha;W)$, which consists of all polynomials $g$ (over the scalar field) such that $g(T)\alpha$ is in $W$. (p.201, Sec. 6.4)

I need some examples of T-conductor

Thanks in Advance

Best Answer

Here's an example. Let $T:\Bbb C^4 \to \Bbb C^4$ be the transformation associated with the Jordan-form matrix $$ \pmatrix{\mu&1&0&0\\0&\mu&0&0\\ 0&0&\lambda&1\\0&0&0&\lambda}. $$ Let $W$ denote the subspace spanned by the vector $e_1 = (1,0,0,0)$. Let $e_1,\dots,e_4$ denote the standard basis of $\Bbb C^n$. We now describe $S_T(\alpha;W)$ for several different vectors $\alpha$:

  1. For any $\alpha \in W$, $S_T(\alpha;W)$ is the set of all polynomials
  2. For $\alpha = e_2$, $S_T(\alpha;W)$ is the set of polynomials divisible by $p(t) = (t-\mu)$
  3. For $\alpha = e_3$, $S_T(\alpha;W)$ is the set of polynomials divisible by $p(t) = (t-\lambda)$
  4. For any $\alpha = (\alpha_1,\alpha_2,\alpha_3,\alpha_4)$ with $\alpha_2 \neq 0$ and $\alpha_4 \neq 0$, $S_T(\alpha;W)$ is the set of polynomials divisible by $(t-\mu)(t-\lambda)^2$.
  5. For any $v \in \Bbb C^4$ and $\alpha = e_1 + v$, $S_T(\alpha;W) = S_T(v;W)$
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