Pick up someone vs Pick someone up?
"Pick up something" or "pick something up"?
Let $M,N$ be $n \times n$ square matrices over an algebraically closed field with the properties that the trace of both matrices coincides along with all powers of the matrix. More specifically, suppose that $\mathrm{Tr}(M^k) = \mathrm{Tr}(N^k)$ for all $1\leq k \leq n$. The following questions about eigenvalues is then natural and I was thinking it would be an application of Cayley-Hamilton but I am having trouble writing out a proof.  How do we show that $M$ and $N$ have the same eigenvalues?  Added (because this question is now target of many duplicates, it should state its hypotheses properly). Assume that all the mentioned values of $k$ are nonzero in the field considered; in other words either the field is of characteristic $0$, or else its prime characteristic $p$ satisfies $p&gt;n$. How do I parse an ISO 8601-formatted date? I need to parse  strings like "2008-09-03T20:56:35.450686Z" into Python's datetime type.  I have found  in the Python standard library, but it is not very convenient.  What is the best way to do this? How to recover/reset forgotten Gnome Keyring Password? Castellation metal loop connector Here's a module with castellated pads attached to an eval board:  (source: ) Looking closely at the wires interfacing with the castellated pads, they're clearly soldered to the eval board. However, they don't look like they're soldered to the module, but rather they're being pressed against the pads with the tension in the wire. It also looks like the free end of the loops are protruding down into a channel cut in the PCB. Do these kind of connectors/metal loops have a name? How can I capture KeyDown event on a WPF Page or UserControl object? I have a Page with a UserControl on it.  If the user presses Esc while anywhere on Page I want to handle.  I thought this would be as easy as hooking up the PreviewKeyDown event, testing for the Esc key, and then handling it.  However, when I placed I breakpoint in the event handler I found it was never getting called.  I thought perhaps the UserControl might be getting hit, so I tried PreviewKeyDown there... same result.  Does anyone know the proper place to test for a KeyDown or PreviewKeyDown on a Page object? Why does an atom remain uncharged after emission of an alpha particle? When an alpha particle is emitted, two protons and two neutrons leave the nucleus but the electrons remain the same in number. Why does the atom remain uncharged although it appears it should have a net charge of $-2e$? Generating a separate pdf file of tables and figures from a latex file? Is it possible to generate a PDF of figures and tables of a paper from LaTeX itself?  I saw Journals and Conferences asking for a separate file of tables and figures.  In  question, questioner proposed a method that uses \usepackage{endfloat} to push all the figures and tables to the last pages and then use a PDF editor to cut the last pages of PDF. This is a solution but not a good one. LaTeX should be able to do this by its own.  What I have in mind is to compile the LaTeX file and it generates a complete PDF output and beside that a separate file that contains all the figures and tables of former PDF. whether it be vs whether it is I brought the sentences from the speech about conversation tips.     They don’t just want to know about news and the weather.   They want to know about you.   So what are three things that you have going on right now, that you can talk about.   Whether it be school, whether it be a personal relationship, whether it be a hobby that you’re involved with.   My questions:     What is the difference in nuance between 'whether it is' and 'whether it be'?   Before posting this question, I googled, and some websites say 'whether it be' is subjunctive. I don't get why it is called 'subjunctive'. Because from what I know about subjunctive, it's supposed to be past of be verb.    If I say 'whether it should be' or whether it would be', is the meaning same? I washed and dried my passport by mistake, do I need a new one? Around two years ago I washed and dried my passport by mistake at the laundry room. My ID, my old US student visa and my stamps from european countries are still readable but the pages are not "flat" anymore and the front cover is kind of erasing (some parts of the Brazil symbol is gone).   In April I'll travel to US for a conference, I'll need to get a new US visa (B1/tourism visa) but I'm worried if my passport is not "good to go" anymore because it was "washed and dried". Apparently there is an electronic chip in it and I don't know if it is still working.  Should I get a new passport or can I save some money and time and use this one? It expires only in 2019. Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$. Story about protagonist finding a world that has died with a museum Number of normal subgroups of a non abelian group of order $21$... CSIR December $2013$ Question is to :  Find the number of Normal subgroups of a nonabelian group $G$ of order $21$ other than $\{e\}$ and $G$.  What I have done so far is :  As $|G|=21=3\cdot7$  we have :  No. of sylow $3$ subgroups $1+3k$ dividing $7$ leaving out possibilities $1$ or $7$  No. of sylow $7$ subgroups $1+7k$ dividing $3$ leaving out only possibilities $1$.  So, we have a unique sylow $7$ subgroup and so it is normal.  I remember somehow that any normal group should come from a normal sylow subgroup or something like that.  S0, I prefer to conclude there is only one Normal subgroup for a non abelian group of order $21$.  Please let me know if this is true and please help me to fill that gap :      I remember somehow that any normal group should come from a normal sylow subgroup or something like that.   Thank you. Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction? I oversaw a high-school mathematics test the other day, and one of the problems was the following     Show, using induction or other means, that $$\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$$   The induction proof is very standard, where the induction step relies on the fact that $\frac{1}{n+1} + \frac{1}{n(n+1)} = \frac{1}{n}$, and I'm sure it's been answered on this site before. However, I got intrigued by the "or other means" part of the question.  I don't know whether the teacher who wrote the test even considered any alternative solutions (he may just have written it so that if anyone has a crazy idea that works out, then they should get a full score for it), but I tried to find one. For instance, we may do the following telescope-ish argument: $$ \frac{1}{1(1+1)} + \frac{1}{2(2+1)} + \cdots + \frac{1}{(n-1)n} + \frac{1}{n(n+1)} + \frac{1}{n+1}\\ = \frac{1}{1(1+1)} + \frac{1}{2(2+1)} + \cdots + \frac{1}{(n-1)n} + \frac1n\\ \vdots\\ = \frac{1}{1(1+1)} + \frac12\\ = 1 $$ However, I feel that this is just an induction proof in disguise (hidden in the vertical dots). If one uses the mechanics of the induction proof to check whether the formula is true for a specific $n$, then one certainly does the exact same calculations as I have done here.  Is there a proof of this fact that clearly does not use induction (or at least hides it better)? The more elementary the better, and the ultimate goal would be to do it within the syllabus of the students taking the test (or at least not far from it). For reference, they should be familiar with the summation formula of arithmetic and geometric series and their derivations (so techniques resembling those would be well within specifications).  If there is a solution using calculus, then the students should be able to integrate elementary trigonometric functions, as well as exponential functions, logarithms and rational functions. They are familiar with integration by parts, substitution and partial fractions.  I welcome more advanced solutions as well, of course. Always return data from last week's Monday thru Sunday Function whose inverse is also its derivative? What are some good examples of a function  $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ and showing that $f^{-1}(x) = f'(x) \implies b-1=\frac{1}{b} \implies b=\phi$ and got $$f(x) = \frac{x^\phi}{\sqrt[\phi]{\phi}}$$ Which seems to work according to WolframAlpha, but I'm having trouble double-checking it. Any other ideas? I've seen references to curried functions in several articles and blogs but I can't find a good explanation (or at least one that makes sense!) I changed the /etc/environment to add more directories to the PATH variable. But looks like $PATH in double quotes were not expanded and I could find any command from the shell and /usr/bin/command-not-found is run.  $ cat /etc/environment  PATH="/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games" PATH="$PATH:/home/user/technical/java/maven/bin" JAVA_HOME="/home/user/technical/java/jdk7"   so from the tty,  $ echo $PATH $PATH:/home/user/technical/java/maven/bin   why did not the expansion happen? Automatically show connected mass storage devices on Windows desktop Locking a vertex group from modification In a mesh there is a group of vertices which should not move after being positioned once. I would like to find a way to lock them in place for the whole duration of the project.   Currently I 'lock' them by hiding them and then sculpting. I imagine it is possible also to create shape keys for the vertices. Or I could write a script which would save position of the vertices.   Is there a better way to do it? Why aren't people voting for questions?