I was reading Linear Algebra by Hoffman Kunze and I found a strange definition
Let $W$ be an invariant subspace for $T\in \mathcal L(V)$ 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 $W$.
The book also added
In the special case $W=\{0\}$ the conductor is called the $T$-annihilator of $\alpha$.
Now, my problem is I don't understand what's going on here. What's the idea behind these two definitions? What's the algebraic (or geometric) picture behind these ideas? I understand the definiton and the intuition behind all the terms used in defining a $T$-conductor. The problem is I just can't connect them together. Please help me to do so.
Also, just after these two definitions mentioned, they proved the lemma
If $W$ is an invariant subspace for $T$, then $W$ is invariant under every polynomial in $T$. Thus, for each $\alpha$ in $V$, the conductor $S(\alpha; W)$ is an ideal in the polynomial algebra $F[X]$.
and went on to say
The unique monic generator of the ideal $S(\alpha;W)$ is also called the $T$-conductor of $\alpha$ into $W$ (the $T$-annihilator in case $W=\{0\}$).
How come these two definitions are equivalent? This is one more connection I can't get. Please help me here.
It seems that there are a couple of questions on MSE that looks like being similar to this one, but actually, they aren't. In these questions, the OP seems to have an idea of what's going on, whereas I don't. One of these questions even had an example, but that didn't help either.
Best Answer
There are really two questions here. The second (why are these two definitions equivalent?) is easier to answer, so I'll answer that one first.
The lemma states that $S(\alpha; W)$ is an ideal in the polynomial algebra $F[X]$. However, the polynomials form a principal ideal domain, which is to say that we can necessarily say that $$ S(\alpha;W) = \{p(x) f(x): p(x) \in F[X]\} $$ for some polynomial $f(x)$ which is called the "generator" of the ideal. Any polynomial for which the above description is valid is called a generator. Notably, we can always choose $f$ to be monic: for any non-zero $k \in F$, we have $$ p(x)f(x) = (k \cdot p(x)) \cdot \frac{f(x)}{k}, $$ so if we take $k$ to be the leading coefficient of $f(x)$, then the above shows us that $f(x)/k$ is also a generator of the same ideal, and because $k$ is the leading coefficient, $f/k$ will be monic.
Now, it is not exactly correct to say that the definitions of $S(\alpha;W)$ and its unique monic generator (which I will denote as $s(\alpha;W)$) are equivalent. The authors do not say that these are equivalent, only that the same term is used for both of these, which is perhaps an unfortunate example of overloaded terminology in mathematics. That being said, it can be argued that $S(\alpha;W)$ and $s(\alpha;W)$ convey "the same information". Indeed, if we are given the set $S(\alpha;W)$, then $s(\alpha;W)$ is just the lowest-degree monic polynomial in $S(\alpha;W)$. If we are given $s(\alpha;W)$, then $S(\alpha;W)$ is just the set of all polynomials of the form $p(x) \cdot s(\alpha;W)$.
As for the idea behind these definitions: these are useful definitions for the derivation of the cyclic decomposition theorem. The second edition of Hoffmann and Kunze gives the following explanation of how these definitions are used.
This is a relatively formal explanation of the importance of $S(\alpha;W)$, so here is a "big picture" perspective. As part of our proof of the cyclic decomposition theorem, we need a convenient way to "verify" whether, for a $T$-invariant subspace $W$ and an arbitrary vector $\alpha$, the invariant subspaces $W$ and $Z(\alpha;T)$ are independent (i.e. $W \cap Z(\alpha;T) = \{0\}$). One nice way to check whether this is the case is to compare the $T$-conductor of $\alpha$ into $W$ with the $T$-annihilator of $\alpha$: these spaces are independent if and only if these polynomials (or equivalently the ideals they generate) are the same. By taking advantage of this, we can start with an arbitrary vector $\beta$, use it to create a new vector $\alpha$, then show that this $\alpha$ has the required property.
To summarize, here is a one-sentence explanation. Together the $T$-annihilator of $\alpha$, the $T$-conductor of $\alpha$ into $W$ describes the (extent of) overlap between $W$ and $Z(\alpha;T)$, the $T$-invariant subspace generated by $\alpha$.