Let A be a near-Jordan block, that is, a matrix obtained from a Jordan block by possibly changing the first column. Prove that no two Jordan blocks in any Jordan canonical form for A have the same eigenvalue.
Oh, I guess I see. So if we change that $\lambda$ to a $\mu$, then the new Jordan form will consist of a single $\mu$ block and a one-lower-multiplicity Jordan block for $\lambda$.
Wait, do you put your $1$'s above the diagonal? That's what I'm doing.
Well it is quite easy to work out the determinant of this new matrix, and I think lamda will only be an eigenvalue if the last element of the first column is 0
Yeah. I'm thinking about just $2\times 2$ matrices.
So then it's dumb, because you either get two distinct eigenvalues or you get one Jordan block with a repeated eigenvalue.
OK, on to $3\times 3$.
The terminology makes me think it still has the same shape, @Gridley, i.e., the first column is $(\text{blah},0,\dots,0)$. Does your exercise not define it more precisely?
Well, here's something to ponder. If the original Jordan block had eigenvalue $\lambda$, we can write the original matrix as $\lambda I + N$, where $N$ is the nilpotent stuff left over ($1$'s above the diagonal).
So $A = \lambda I + N + C$, where $C$ is a nonzero first column.
For $\mu$ to be an eigenvalue of this, $\mu-lambda$ must be an eigenvalue of $N+C$. Can you play with that?
So I guess that's what you need to think about. You get the original eigenvalues by adding $\lambda$ to those. So it suffices to understand the $n$th degree polynomial with the entries of $C$ as coefficients. (I don't know if you've seen rational canonical form. But this is related to that.)
So it reduces to understanding the Jordan form of $N+C$ and seeing no eigenvalue repeats in its Jordan blocks. (That's not a deep reduction we've made.)
But now I don't see why I should believe that.
Well, when you figure this out, please let me know :)
@TedShifrin I think I asked this yesterday, but: can it happen that in a manifold a geodesic doesn't exist (so not complete) between two points but a locally distance minimizing path does? In $\Bbb R^2 - 0$ for example neither exists between a pair of antipodal points.
I don't see what pointwise control over $|f-g|$ controls the difference in oscillations. We can cook up an example where $g$ oscillates off-period from $f$.
If you just combine the infs/sups, then $$\inf_{\delta>0}\sup_{I_\delta(x)}[{|f(y)-f(z)|}-{|g(y)-g(z)|}].$$ Some manipulations give \begin{align*} |f(y)-f(z)|-|g(y)-g(z)|&\le |f(y)|-|g(y)|+|f(z)|-|g(z)|\\ &=|f(y)-g(y)|+|f(z)-g(z)|\le 2\epsilon, \end{align*} so that $$|\omega_f(x)-\omega_g(x)|\le \left|\inf_{\delta>0}\sup_{I_\delta(x)} 2\epsilon\right|=2\epsilon.$$
If we somehow make it past Nov 8, let me know when you get to Athens. We'll have to get a group of past students together. By then I should be finished with applying to grad schools, which will be a relief! I'm still trying to choose where to apply to.
I like ones with mysticism. Like "limpia," "smudging." "energy healing," etc. But if they talk I have to reduce volume so as to not hear their nonsense.
It's not really asking you anything, it's just defining a relation on the set of integers.
@MikeMiller did you mean the asmrs? Dmitri's got a few where he doesn't do weirdo role-playing or nonsense talking stuff, but uses household materials to produce nice noise. I like those more.
none of them ever really worked for my insomnia anyway, so I stopped watching
a student, this afternoon: "this set is open, hence it is not closed: this is why [...]"
