Recent content by NotEuler

Thanks Hill and fresh_42. Perhaps I was again confusing vectors with their coordinates, like you mentioned very early on. I'll try to explain where my original question came from, because I am not sure I phrased the question very well. My friend was working on an exercise that went more or less...

Ok, I am making progress. But now I am stuck with one thing: so the diagonal matrix in post 2 is symmetric. But once it is transformed into cartesian coordinates like fresh_42 explained, is that transformed result necessarily symmetric? And if it is not, then is a right eigenvector of the matrix...

Ok, I found my mistake and managed to replicate what you describe here. That diagram is really really helpful for my understanding of what is going on! There are still things I don't understand, but I will have to think about how I can articulate those things. I don't think I understand them...

Many thanks for your time and effort! I am really learning a lot from this. I thought that is what I did earlier in my 'experiments', but I must have messed up somewhere. I shall give it another go when I get the chance and will let you know if I succeed or not! Thanks again.

I tried to do this but I don't think I understand it yet. So should the basis transformation matrix be formed of the eigenvectors a1 and a2? This is what I tried to do but I don't think it worked. But probably I am writing complete nonsense...

True. I didn't really think of that. And in that case, I should rephrase my question as 'Can I be sure a _non-zero_ matrix exists that has those eigenvectors?'

Thanks very much for all this. I shall think on it and hopefully I'll eventually get my head around it!

Yes, I think I partially get this. I should not be thinking of the vectors as entities tied to one specific coordinate system, but more generally in any basis? And its coordinates will vary depending on which basis I am working in? So with your presentation and Hill's advice I can construct a...

Ok, thanks to both of you. I think I'll have to sit down with pen&paper and maybe a book or two to figure this out...!

Thanks! But I'm not sure if this is the same question? You are talking about two eigenvectors to two different eigenvalues. My question was about the left and right eigenvector, both to the same eigenvalue.

I tried to find the answer to this but so far no luck. I have been thinking of the following: I generate two random vectors of the same length and assign one of them as the right eigenvector and the other as the left eigenvector. Can I be sure a matrix exists that has those eigenvectors?

Thanks pasmith, looks great. One question: Is it always valid to take the limit of the two R:s in the last integral simultaneously? What I mean is that in the integral ##\int_0^{\infty } \left(\int_x^{\infty } f(y) \, dy\right) \, dx## it seems like the two infinities are 'separate'. In your...

Ah yes, I think I see at least partly. If I write ##F(x)=-\int_x^{\infty } f(y) \, dy##, then ##\int_0^{\infty } x f(x) \, dx = \int_0^{\infty } x F'(x) \, dx = [xF(x)]_0^∞ - \int_0^{\infty } F(x) \, dx##. ##[xF(x)]_0^∞## goes to 0 at the lower limit if ##F(x)## converges, but I am not quite...

Now I am wondering if my counterexample really is a counterexample. ##\int_0^{\infty } \left(\int_x^{\infty } f(y) \, dy\right) \, dx## and ##\int_0^{\infty } x f(x) \, dx## both diverge with ##f(y) = \frac{1}{(y+1)^2}##, so I suppose from that perspective the conjecture still holds? Either...

Here's an example of where my guess works: ##f(y) = \frac{1}{(y+1)^3}## Then: ##\int_0^{\infty } \left(\int_x^{\infty } f(y) \, dy\right) \, dx = \int_0^{\infty } \ \frac{1}{2 (x+1)^2} \, dx = 1/2 ## And also: ##\int_0^{\infty } x f(x) \, dx = 1/2## So my guess gives the correct answer with...