I am hoping someone can help me begin to address my deficiencies in understanding this problem. On Math Stack Exchange, the OP is asking how to find eigenvalues and eigenvectors of block Toeplitz matrices. I have a similar problem I wish to solve, but I would like to understand their work first, so my question is this:
How did the OP go from, T, a 2N-by-2N matrix, represented as the sum of tensor products (the Pauli matrix step) to T as the sum of (2-by-2) matrices after plugging in the eigenvalues of each (N-by-N) sub-block of T?
I understand that the eigenvalues of the tridiagonal Toeplitz matrices fill each of the elements of the 2-by-2 representation of T, and also that the eigenvectors of tridiagonal N-by-N Toeplitz matrices (e.g., A, B, etc) are all the same, and thus simultaneously diagonalize all blocks of T(2N-by-2N).
Thanks for the help!
My boss asked what I was doing with my measurements and calculator - I proudly showed her. The new bulbs are a perfect “X” in the dining room. Thank you to the math teachers of Lake High School. 🤘
New Mayor of NYC Eric Adams gave a speech before todays Knicks-Hornets game and featured a “hello fellow basketball fans” level quote
>"When the civil rights game battle (corrected) was on the line, Dr. King wanted the ball in his hands," Mayor Eric Adams said before tipoff of today's Knicks-Hornets game.
> https://twitter.com/bobsaietta/status/1483139138111033351?s=21
> Tweet with correction + a fan voicing his displeasure of the speech
> https://twitter.com/bobsaietta/status/1483146244612669444?s=21
Basically a theorem that says “all but some number of cases” satisfies the theorem
I don't think I've ever seen this movie. It looks like a typical college humor comedy from the American Pie era of fine comedy. I do really like Steve Zahn and Jack Black, but the reviews seem terrible. Is this a cultural touchstone I need to see or is it okay that I missed it?
I'm pretty sure I've heard the name before and that makes me think it was popular, but the bad ratings worry me.
Or even better, wasn’t Mike a big fan of golden girls too? And now Betty White is gone as well. It’s all very very suspicious to say the least
Howdy, y’all! First time poster here. :)
So, I learned about the FToA this semester, in my Pre-Calc class, but my textbook honestly did kind of a bad job explaining it. (Didn’t even have any sort of proof for it, for one thing.[Edit: Turned out to be good reasons for this, lmao.]) This video helped clear it up a bit for me, but I’m still having trouble understanding it. Like, what is it about this theorem, in particular, that makes it “the link between algebra and geometry” (to paraphrase that vid, iirc), what is is that makes this theorem in particular so important? I get why the Fundamental Theorem of Calculus gets the title, but idk, I feel like I’m just not quite getting or grasping something here, and not understanding why this is so important to algebra.
Edit: I appreciate all the responses! Y'all definitely have given me a better perspective on it, and what I should further study to better understand it.
It me, I'm the 8th grade math teacher. It's always bothered me that we can't easily "see" the solutions that the FTA tells us to expect for polynomial equations if those solutions are in the complex numbers rather than the real numbers. So when I introduce the FTA to students, I usually talk about it in terms of real solutions and how it tells us to expect up to some number of real solutions depending on the degree of the polynomial. This video is my attempt to connect that idea to the actual statement of the FTA relating polynomial degree to complex solutions, and to visualize where the real solutions "go" if we, for example, vertically shift a third-degree polynomial from three to two to one real solution.
Perhaps not at the level of most of the commenters here, but I'd love to know if I'm making any egregious missteps in how I discussed the math.
I also said at least one thing that, after the fact, I realized I wasn't really sure about. Would you say a mapping of a polynomial function's complex inputs to its complex outputs makes a two-dimensional surface in four-dimensional space?
