Find 2 x 2 matrices E and F such that EF = - FE? The questions specifies that EF cannot equal 0. I was just wondering if there was an intuitive way to find an example of this relationship, without having to define each matrix, multiply them out and come up with a few conditions that need to be satisified. Could eiegenvectors and eigenvalues be used? Thanks
Matrices - Conditions for $AB+BA=0$ The Problem Let $A$ be the matrix $\bigl(\begin{smallmatrix}a&amp;b\\c&amp;d\end{smallmatrix} \bigr)$, where no one of $a,b,c,d$ is $0$. Let $B$ be a $2\times 2$ matrix such that $AB+BA=\bigl(\begin{smallmatrix} 0&amp;0\\ 0&amp;0 \end{smallmatrix} \bigr)$. Show that either   $a+d=0$, in which case the general solution for $B$ depends on 2 parameters, or $ad-bc=0$, in which case the general solution for $B$ depends on one parameter.   (this is question 22 of the last matrix exercise of Further Pure Mathematics by Bostock et al.)  Comments Writing $B=\bigl(\begin{smallmatrix} e&amp;f\\ g&amp;h \end{smallmatrix} \bigr)$ and multiplying out I get that    $(a+d)(f+g)+(b+c)(e+h)=0$ $ae+bg+cf+dh=0$   but I am unable to get the required restrictions on $a,b,c,d$. Is there a quick way of doing the problem that doesn't require manual computation? I thought of considering invertible and non-invertible cases but couldn't get anywhere. Help would be much appreciated.
SQlite: select into? I have a database with millions of phone numbers with free-for-all formatting. Ie, the UI does not enforce any constraints and the users are typing in whatever they want.  What I'm looking for is a Java API that can make a best-effort to convert these into a consistent format. Ideally, the API would take the free text value and a country code and produce a valid international phone number or throw an exception.  For example, a phone number in the system might look like any of the following:  (555) 478-1123 555-478-1123     555.478.1123 5554781123   Given the country of US, the API would produce the value "+1 (555) 478-1123" for all these. The exact format does not matter, as long as it's consistent.  There are also numbers in the system without area codes, such as "478-1123". In that case, I would expect a NoAreaCodeException, or something similar.  There could also be data such as "abc", which should also throw exceptions.  Of course, there are countless variations of the examples I have posted, as well as the enormous complication of international phone numbers, which have quite complicated validation rules. This is why I would not consider rolling my own.  Has anyone seen such an API? Given the following code, what does the if __name__ == "__main__": do?  # Threading example import time, thread  def myfunction(string, sleeptime, lock, *args):     while True:         lock.acquire()         time.sleep(sleeptime)         lock.release()         time.sleep(sleeptime)  if __name__ == "__main__":     lock = thread.allocate_lock()     thread.start_new_thread(myfunction, ("Thread #: 1", 2, lock))     thread.start_new_thread(myfunction, ("Thread #: 2", 2, lock)) Navigation to section/subsection from Beamer headline titles Beamer infolines themes, like CambridgeUS, insert the titles of the current section and subsection in the headline. These titles appear to be clickable (the screen pointer changes as if they were so) but nothing happens when one clicks them.   Is there a way to link the headline titles to the first frame of the corresponding section or subsection? Something like this already works on the theme footline: clicking on the presentation title takes you to the title page.   Edit: MWE  \documentclass{beamer} \usetheme{CambridgeUS}  \setbeamertemplate{navigation symbols}{} \title{Click on Headline} \author{A.~U.~Thor} \institute{X-University} \date{2015} \begin{document} \begin{frame}   \titlepage \end{frame} \begin{frame}[t]{Contents}   \tableofcontents[subsectionstyle=hide]  \end{frame} \section{Section 1} \begin{frame}{Section 1}   \centering \Large Section 1 \end{frame} \begin{frame}[t]{Section 1: Contents}   \tableofcontents[currentsection,hideothersubsections] \end{frame} \subsection{Subsection 1.1} \begin{frame}{Frame 1.1.1}   A frame of subsection 1.1 \end{frame} \begin{frame}{Frame 1.1.2}   Another frame of subsection 1.1 \end{frame}   \subsection{Subsection 1.2} \begin{frame}{Frame 1.2.1}   A frame of subsection 1.2 \end{frame} \begin{frame}{Frame 1.2.2}   Another of subsection 1.2 \end{frame} \section{Section 2} \begin{frame}{Section 2}   \centering \Large Section 2 \end{frame} \begin{frame}[t]{Section 2: Contents}   \tableofcontents[currentsection,hideothersubsections] \end{frame} \subsection{Subsection 2.1} \begin{frame}{Frame 2.1.1}   A frame of subsection 2.1 \end{frame} \end{document} Who turned on the lightsaber? Spoilers for The Last Jedi:      Just before Snoke is slain,   he narrates Kylo Ren's thoughts/feelings (saying things like "he turns the lightsaber towards his enemy" and so on) we see the Skywalker lightsaber being manipulated through the Force to point towards and then slay Snoke. My question is, do we know for sure who was using to Force to use the Skywalker lightsaber in this scene?   It could have been   A) Kylo Ren was turning igniting the Skywalker lightsaber with the Force. Snoke assumed the thoughts/feelings he was sensing were directed at Rey but they weren't.   Or... B) Snoke was so preoccupied with Kylo's thoughts that he wasn't paying any attention to Rey's thoughts. Rey takes advantage of this and uses the Force to turn and ignite the Skywalker lightsaber to slay Snoke. This option is reinforced by the fact that  1) the Skywalker lightsaber then zooms into Rey's hand, not Kylo's.  2) after the ensuing scuffle, Kylo Ren reveals that he hasn't switched sides; he is still a believer in the First Order.  3) Kylo Ren even tells Hux that Rey slew Snoke. "Startup Applications" not working Possible (but not serious) glitch in close/reopen I just had one of my questions closed. (Hey - it happens to everyone) When I clicked on reopen - lo and behold - it worked. I cast a reopen vote!  Is this a glitch? I know we need 4 more votes anyway but still...  And no  tag please :) Showing two ring homomorphisms that agree on the integers must agree on the rationals I have two ring homomorphisms $f,g\colon \mathbb{Q}\to X$. I know that $f=g$ on the integers. How can I show that $f$ and $g$ agree on the rationals?   My attempt:  let $x,y \in \mathbb{Z}$, $y\neq 0$.  $$\begin{align*} f(x/y)&amp;=f(x)f(1/y) &amp;&amp;\text{because }f\text{ is a ring homomorphism}\\  &amp;=g(x)f(1/y) &amp;&amp;\text{because }f\text{ and }g\text{ agree on the integers} \end{align*}$$  I do not know how to rewrite $f(1/y)$ so that I can replace it with $g(1/y)$. Any hints will be much appreciated. Thanks. What's the difference between Fermi Energy and Fermi Level? I'm a bit confused about the difference between these two concepts. According to Wikipedia the  and  are closely related concepts. From my understanding, the Fermi energy is the highest occupied energy level of a system in absolute zero? Is that correct? Then what's the difference between Fermi energy and Fermi level? Why there's no article with 'Mexican' and 'an' with American in sentence: "She’s Mexican. But she’s married to an American."   (Pimsleur English course) The external Flash is not showing up in the configuration Menu when using Pixel TF-321 hot shoe. Is this the wrong adapter for that camera, as there is no X connection? I want to trigger a studio flash via sync connection. Can you recommend me a working hot shoe? Thanks! How long can a flag remain active? I have flagged the following question as a duplicate.    Now I want to know: after how long time will it be stated as active?  I do not have problem with anything here. I just want to know after how long time the request (of a duplicate question) will automatically be rejected or approved.  OK, I got the things, so it will remain active until moderators will have a look at it...  But don't you think we should have a limit that after a certain number of days this kind of request will be dropped (or cancelled)? Are there rules about using "that" to join two clauses? He will understand that I was not joking.   He will understand I was not joking.  Which of the sentences is correct? Are there any specific rules about the use of &quot;that&quot; in the sentences I reported as example? Why is my camera limited to a shutter speed of 1/250th when the flash is up? Sometimes in bright backlight and with large aperture (f2.8) I want flash at high shutter speeds.  But when I pop the flash up on D7000 it caps the shutter speed to 1/250th, which overexposes the shots and makes everything look like a nuclear bomb just went off.  What's the logic behind the limiting of the shutter speed when using flash? Is there anything I can do to get flash  + faster shutter speeds? Comparing complex numbers If $a+ib$, $c+id$, $e+if$ are three complex numbers, than can we tell which one is greater or smaller between them? If yes, then how and if no then why not?   Can somebody give explanation on this.... I will be grateful to him. Negative energy levels in the diagram for a hydrogen atom If $(A-\lambda{I})$ is $\lambda$-equivalent to $(B-\lambda{I})$ then $A$ is similar to $B$ How can I iterate, loop, process, or otherwise do something, with each element from a comma separated list of things?  This question       , and the are already lots of very good answers everywhere. However I think it would also be valuable to have a single reference point where all such alternative techniques are explained in detail, with their advantages and disadvantages discussed.  All of plain-TeX, LaTeX, LaTeX3, LuaTeX, Context, etoolbox, pgffor, etc. answers are invited. Please consider explaining, for example:   How does the command/method deal with whitespace around items? How does the command/method deal with empty/missing items? (e.g. a,b,,c) How does the command/method deal with a trailing comma? Can the command/method work of a list stored on a macro? Is each iteration evaluated globally or on a local group? (i.e., if a command is defined during an iteration, does the command survive the loop?) Do items get expanded or somehow mangled before being processed? What about lists where items are key=value or some other kind of pairs? Does the method work for lists separated with something other than a comma? Any other thing I should know before deciding to use this method? I have an Excel file containing data collected over many years as latitude and longitude coordinates for archaeological sites. How can I import it into QGIS as a new layer? Draw a vertical line over the entries of a column in an array I'd like to draw vertical/horizontal lines over the entries of a column/row in an array. I find the method to draw horizontal lines at . But when I used the same way to draw the vertical lines. It didn't look well. My codes are:  \usepackage{tikz} \usetikzlibrary{calc} \newcommand{\tikzmark}[1]{\tikz[overlay,remember picture] \node (#1) {};} \newcommand{\DrawLine}[3][]{% \begin{tikzpicture}[overlay,remember picture]     \draw [#1] ($(#2)+(0,0.6ex)$) -- ($(#3)+(0,0.6ex)$); \end{tikzpicture}% }%  \begin{displaymath} \left[ \begin{array}{cccc} \tikzmark{11}{15} &amp; 0 &amp; \tikzmark{13}{0} &amp;0\tikzmark{14} \\ 0 &amp; 50 &amp; 20 &amp; 25 \\ 35 &amp; 5 &amp; 0 &amp; 10\\ \tikzmark{41}{0} &amp; 65 &amp; \tikzmark{43}{50} &amp; 65 \end{array} \right]. \DrawLine[black,thick]{11}{14} \DrawLine[black,thick]{11}{41} \DrawLine[black,thick]{13}{43} \end{displaymath}   The result looks like     But the result I want is     Does anyone know how to fix this? Thank you.