I had a linear algebra test today, and there were 6 questions which were of the type "say true with proof, or false with counterexamples". The sad part was (as far as I figured out) only one of them was true, and the other five needed counterexamples. These 5 were (as far as I can recall)
Two $3\times 3$ matrices on $\mathbb C$ with the same minimal polynomial must be similar.
Two $3\times 3$ nilpotent matrices must be similar.
Two $4\times 4$ nilpotent matrices with same minimal and characteristic polynomial must be similar.
Product of two nilpotent matrices is nilpotent.
For any operator $T$ on $V$, we have $V=\text{range }T\oplus \text{ker }T$.
(Calm down, I'm not asking these five questions together)
Now, although (I believe) I could correctly arrive at these counterexamples, it made me completely exhausted. In other math courses, when we were constructing counterexamples, life didn't seem to be this difficult as I was mostly familiar with the properties of the real numbers or functions $f:\mathbb R\to \mathbb R$, or sequences of real numbers. I had very concrete idea of how they worked, and doing some brute force calculations weren't that difficult. In here however, things seemed really hard- I had no idea how to quickly guess a matrix with a desired minimal or characteristic polynomial, I didn't know how to construct "the simplest" $4\times 4$ nilpotent matrix (except that upper triangular matrices with $0$ diagonal is nilpotent), and matrices cannot be quickly multiplied with brute force like real numbers.
This got me thinking. I want to know what are some of the tricks and properties, or some of the more famous candidate operators and matrices that we should remember to construct these quick counterexamples.
I understand that my question sounds a little vague. So, here's some more explanation. In real analysis, when we needed to find pathological examples of continuous (/discontinuous) functions, the Dirichlet function was more often than not called for aid. Similarly, in sequences, $\{(-1)^n\}_{n=1}^\infty$ was often our first guess; in cases concerning metric spaces, the Discrete Metric Space was often the first choice for a pathological example. So, my question is whether there are any such "first guess" matrices, or any candidate matrices designed specially to bend the rules that we may use. In other words, what are the properties that I should remember (considering the fact that there are too many nitty-gritties to remember all of them) to have sufficient hints to construct counterexamples?
Although I've accepted an answer, please feel free to still continue to pour interesting answers into this post, and recieve upvotes (at least) from me.
Best Answer
There's an approach to these problems that I largely attribute to Artin's Algebra which is: when evaluating whether a claim is true or not, try to evaluate with some very simple representatives. Frequently you'll find a counterexample. (Alternatively, if true, you can frequently bootstrap a general proof from a simple one.) Diagonal matrices are of course easiest, but beyond that
(i.) upper triangular matrices
(ii.) Hermitian matrices are very easy to work with and have properties you can easily exploit.
(iii.) block diagonal matrices are also quite nice.
This motivates a very useful technique: work out some $2\times 2$ and perhaps $3\times 3$ examples then use them in simple block structures to create more complicated examples. In general block diagonal matrices of the form
$A= \left[\begin{matrix}Y & \mathbf 0\\\mathbf 0 & Z\end{matrix}\right]$
where ($Y$ and $Z$ are square) are extremely easy to work with. There are underlying interpretations in terms of direct sums (and direct products) that can be helpful. I trust you know that the characteristic polynomial splits along the blocks, i.e.
$\det\big(\lambda I -A\big)=\det\big(\lambda I -Y\big)\cdot \det\big(\lambda I -Z\big)$
(iv.) involutions and idempotence also have very nice properties, useful elsewhere
For this post, every nilpotent matrix is similar to a strictly upper triangular matrix, so use them as a representatives in $2-4$ (and even 5).
The $2\times 2$ case is simplest so start with that. Here we only have representatives $\mathbf 0$ and $N = \left[\begin{matrix}0 & \alpha\\0 & 0\end{matrix}\right]$ for some $\alpha \neq 0$. These are obviously not similar, justification: (a) the conjugacy class of the zero matrix only contains the zero matrix or (b) they have different ranks so can't be similar.
(1) requires some tinkering though I'd attack this in terms of either diagonal matrices or use (iii).
(2) Consider re-using the the $2\times 2$ case (and trivial $1\times 1$ case) in blocks $\left[\begin{matrix}\mathbf 0 & \mathbf 0\\\mathbf 0 & 0\end{matrix}\right]$ and $\left[\begin{matrix}N & \mathbf 0\\\mathbf 0 & 0\end{matrix}\right]$. Same rationale for why they can't be similar.
(3) since $4=2\cdot 2$, consider the $2\times 2$ case, twice, via a block matrix $A:= \left[\begin{matrix}N & \mathbf 0\\\mathbf 0 & N\end{matrix}\right]$ and $A':= \left[\begin{matrix}N & \mathbf 0\\\mathbf 0 & \mathbf 0\end{matrix}\right]$
$\text{rank } A =2 \gt 1 =\text{rank } A'$ so they aren't similar but they have the same minimal and characteristic polynomials.
note: you said
This is one of the advantages of the block diagonal structure. You learn the minimal (and characteristic) polynomials for simple (e.g. $2\times 2$) cases then infer the result for some more complicated matrix living in a higher dimensional vector space. i.e.
$p\big(A\big):= \left[\begin{matrix}p\big(N\big) & \mathbf 0\\\mathbf 0 & p\big(N\big)\end{matrix}\right]$
so it's immediate that $A$ has the same minimal polynomial as $N$. And you can check that $A'$ has same min polynomial -- anything of lower degree wouldn't annihilate $N$ on its leading block.
(4) Make using of Hermicity so $Z: =N^*N$ is a product of two nilpotent matrices but e.g. Hermitian matrices are diagonalizable and the only diagonalizable nilpotent matrix is the zero matrix, but $\text{rank}\big(N^*N\big)=\text{rank}\big(N\big)=1\gt 0$, etc. There are many ways to finish, but the big idea is you can convert an awkward matrix into a diagonalizable one via Hermicity.
(5) a simple approach is to recognize that you have been getting hit with nilpotent matrices so use them. Again with $2\times 2$ case we have $N(N) =N^2 = \mathbf 0$ but $N\neq \mathbf 0$ so $N$ kills $N\cdot V \neq \mathbf 0$ which tells you something about $\ker N \cap \text{im } N$.