Applications of the annihilator from linear algebra

Question

I am currently assisting a course for future teachers at university level for a joint education in maths and physics in Germany and I have a question regarding the use and possible application of the following result.

We defined the annihilator for an element A of a K-algebra $\mathcal A$ as the set of polynomials over the field K, that return zero, i.e.

$$Ann(A):=\{f\in K[x]: \tilde f(A)=0\}$$

The tilda indicates the evaluation-homomorphism. We proved that any such f can be decomposed into a unique normed polynomial h and some rest, such that every $f = g\cdot h$. This can be used to show that a nilpotent matrix of order k, has $M_A(x)=x^k$ as minimal polynomial.

My problem: it is an interesting tool to find the minimal polynomial of say a real valued matrix. But I do not see the relevance for future teachers as this seems to be "just" a technical result without any applications that one could explain to someone without extensive mathematical education. I do know, that the subspace of linear functions with the same property is a subspace of the dual space.

So, are there any nice applications for this method in the realm of euclidean geometry? or physics? or maths?

If you feel this is not the right place to ask such an involved question, feel free to suggest an alternative. I have posted this very question to math.stackexchange, too.

Answer 1

The method of annihilators from constant coefficient ODEs is interesting. If $L = P(D) \in \mathbb{R}[D]$ where $D = d/dt$ and $deg(P)=n$ then $L[y]=0$ defines an $n$-th order homogeneous ODE. In this case, $Ann(L)$ is a finite dimensional subspace of the infinite dimensional space of smooth functions on $\mathbb{R}$. So, one interesting thing is just that annihilators give a formalism to characterize solution sets as they relate to the structure of an operator. This alone is not terribly interesting, the method of annihilators typically refers to the following mathematical slight of hand: given a nonhomogeneous $n$-th order ODE $L[y] = g$ we can apply the method of annihilators if there exists $A \in \mathbb{R}[D]$ with $deg(A)=k$ for which $A[g]=0$. Then, $AL[y] = A[g] = 0$. By the theory of ODEs, there exists a fundamental solution set for $AL[y]=0$ say $y_1,y_2,\dots, y_n, y_{n+1}, \dots , y_{n+k}$. Moreover, without loss of generality we may suppose $y_1,y_2,\dots, y_n$ is a fundamental solution set of $L[y]=0$ since solutions of $L[y]=0$ are also solutions of $AL[y]=0$. In fact, we can argue $y=\displaystyle\sum_{j=1}^{n+k} c_jy_j$ serves as the general solution of $L[y]=g$.

For example, $y''-y = e^t$ has $L = D^2-1$ and $A=D-1$ gives $A[e^t]=0$. Note, $AL = (D-1)(D^2-1) = (D+1)(D-1)^2$ thus $AL[y]=0$ has solution $y = c_1e^{-t}+c_2e^t+c_3te^t$. Observe $L[y] = e^t$ gives $$ (D^2-1)(c_1e^{-t}+c_2e^t+c_3te^t) = e^t $$ hence $$ c_3D^2[te^t] - c_3te^t = e^t $$ or $$ c_3(te^t+2e^t) - c_3te^t = e^t $$ consequently, $$ 2c_3e^t = e^t $$ and we find $c_3 = \frac{1}{2}$ thus $y = c_1e^{-t}+c_2e^t+ \frac{1}{2}te^t$ is the general solution of $y''-y=e^t$

Notice the nonhomogeneous problem can be viewed as a finite dimensional linear algebra problem since $ann(AL)$ is an $(n+k)$-dimensional vector space by the existence and uniqueness theory for constant coefficient ODEs. I think this is fairly removed from direct matrix math. Of course, pick a basis, and we can be right back to the matrix.

Beyond this, the concept of annihilators play a large role in the study of differential forms and integral submanifolds. But, you might be looking for a very different sort of example, so I'll stop here.

Answer 2

I'm not a theoretical math type, more of a technical guy. But I'm used to seeing the annihilator method and operator discussed in ODE courses. For example pages 193-195 of Spiegel, third edition Applied Differential Equations.

If there are other knuckle-draggers like I in your course (since the physics slant), perhaps they will find such a discussion (having to do with characteristic equation of familiar second order ODE with constant coefficients) easier to grok than the holomorphism and set notation above.