If two matrices $A$ and $B$ commute with each other, why would they share some eigenvector?  Does that mean that an eigenvector for $A$ is also an eigenvector for $B$ and vice-versa?
I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement?
Say I have a code to draw a flow chart, I want now a arrow from node 3 to node 1, how can I do that?  1&lt;--| |   | 2   | |   | 3----   MWE  \documentclass[class=article,border=0pt]{standalone}  \usepackage{tikz} \usetikzlibrary{shapes,arrows}  % Define block styles \tikzset{     block/.style={rectangle, draw, line width=0.5mm, black, text width=5em, text centered,                   rounded corners, minimum height=2em},     line/.style={draw, -latex} }% &lt;- if you insist in using this in the document add this % here. \begin{document} \begin{tikzpicture}[node distance = 1cm, auto]     % Place nodes     \node [block] (BLOCK1) {a};     \node [block, below of=BLOCK1] (BLOCK2) {b};     \node [block, below of=BLOCK2, node distance=1cm] (BLOCK3) {c};     % Draw edges     \path [line] (BLOCK1) -- (BLOCK2);     \path [line] (BLOCK2) -- (BLOCK3); \end{tikzpicture} \end{document} Need help understanding "getbits()" method in Chapter 2 of K&R C In chapter 2, the section on bitwise operators (section 2.9), I'm having trouble understanding how one of the sample methods works.  Here's the method provided:  unsigned int getbits(unsigned int x, int p, int n) {     return (x &gt;&gt; (p + 1 - n)) &amp; ~(~0 &lt;&lt; n); }   The idea is that, for the given number x, it will return the n bits starting at position p, counting from the right (with the farthest right bit being position 0). Given the following main() method:  int main(void) {     int x = 0xF994, p = 4, n = 3;     int z = getbits(x, p, n);     printf("getbits(%u (%x), %d, %d) = %u (%X)\n", x, x, p, n, z, z);      return 0; }   The output is:     getbits(63892 (f994), 4, 3) = 5 (5)   I get portions of this, but am having trouble with the "big picture," mostly because of the bits (no pun intended) that I don't understand.  The part I'm specifically having issues with is the complements piece: ~(~0 &lt;&lt; n). I think I get the first part, dealing with x; it's this part (and then the mask) that I'm struggling with -- and how it all comes together to actually retrieve those bits. (Which I've verified it is doing, both with code and checking my results using calc.exe -- thank God it has a binary view!)  Any help? Dropbox: ext4 isn't ext4 Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$   Obviously $(2,1)$ and $(1,2)$ are two answers. But I was unable to manipulate the equation algebraically giving a useful form for finding all other possible answers! I also tried to view it as a quadratic equation in terms of $x$, but forming the Delta of that equation didn't help much other than if $y=1$ then certainly Delta would be a square and equation would have answer for $x$... How to evaluate potential advisers on grounds other than their research/publications? While accepting an offer to grad school, one is basically entering into a lasting relationship with one's adviser - most likely, someone whom the applicant has never met before, and the only exposure has been through the potential adviser's website/publications. It is in the interest of both parties to ensure (to the greatest possible extent) that there are no personal/professional traits of either that hamper the formation of a pleasant working relationship - no-one would want to go through the ordeal of having to change advisers midway!  While the faculty has a chance to have a good look at the applicant's profile as well as his motivations (through his grades and SOP), the applicant doesn't have a similar opportunity. So, I'm interested to know what parameters can be used to gauge a potential fit. I've thought of the following:   Past students:  Did anyone ever drop out/change advisers midway, and if so, for what reasons? These would be a bit hard to find though, as I don't expect the faculty concerned would list them on their website. It would be great if anyone could let me know how to find out the list of incoming students to a department for any year. Publication rate, taking into account the venues where they were accepted. Time taken to graduate - though I accept this is more dependent on the student, a median figure should be telling ... What they did post Ph.D. - did anyone get tenure if they went into academia, or is almost everyone unable to break out of being a post-doc?  Tenure status: I'm a bit unsure about this, so wanted the community's opinions about it. Just so that I'm clear, I'm only trying to calibrate the applicant's expectations about the working style of his potential adviser - and hence need to know to what extent are the following "typical" assumptions valid.  Tenured professor  A full professor is more likely to get grant funding, hence less time spent on TAship - but could also mean less time/effort spent on interactions with students (either being busy with other projects/talks, or due to more commitments to family at that age).  Tenure-track faculty (Assistant profs)  More likely to be young and energetic, and could translate to more time spent on one-to-one discussions with grad students - but funding may prove to be an issue, and may have to be on TA for a longer period.   What other factors would be relevant in this matter, and to what extent am I correct/incorrect in either the factors considered, or for undertaking this exercise at all? Redirecting stdout to a file you don't have write permission on When you attempt to modify a file without having write permissions on it, you get an error:  &gt; touch /tmp/foo &amp;&amp; sudo chown root /tmp/foo &gt; echo test &gt; /tmp/foo zsh: permission denied: /tmp/foo   Sudoing doesn't help, because it runs the command as root, but the shell handles redirecting stdout and opens the file as you anyway:  &gt; sudo echo test &gt; /tmp/foo zsh: permission denied: /tmp/foo   Is there an easy way to redirect stdout to a file you don't have permission to write to, besides opening a shell as root and manipulating the file that way?  &gt; sudo su # echo test &gt; /tmp/foo The pronunciation of the word "a" Most times we hear that the letter A is pronounced as [ə], but sometimes I hear that it is pronounced as [eɪ] in American English. Such as the following sentence: From VOA Special English:  She says governments must make that goal of a continental free trade area a reality.  The first A is pronounced as [eɪ], the second A  is pronounced as [ə]. Why? Does it have some regularity in it? How can I read off the fact that gravity is associated with spin-2 particles from the Einstein-Hilbert action? I have often heard that the gravitational field has spin $2$. How can I read the spin of the field from the Einstein-Hilbert action  $$S=\int \! \mathrm{d}^4x \,\sqrt{|g|} \, \mathcal{R} \, \, \, ?$$ When asking about , people sometimes refer to the strict aliasing rule. What are they talking about? Are proofs by contradiction really logical? Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction.   My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"?  That, to me, seems the exact opposite of logical. It sounds like we say "so, I'll have a really big problem if this thing isn't true, so out of convenience, I am just going to act like it's true". What is the single most influential book every designer should read? What is the single most influential book every designer should read? Featured On Meta section doesn't list MSE posts on meta sites My father in law is travelling to France, for 4 days, (01/07-04/07). Is there a minimum duration for which the travel insurance needed to apply for a Schengen visa must be valid? In other words, can his travel insurance cover just the duration of the trip, or should it last longer? How do you display a photo map tip in QGIS 3 on Windows?  As a new user I'm not sure if this is my lack of experience or a bug, so I've tried to simplify it as far as I can:  This is the contents of the Layer Display properties.    &lt;div style="width:400;height:400"&gt; &lt;p&gt;test&lt;/p&gt; &lt;div&gt;&lt;img src="C:\Local\aa.jpg" width="350" /&gt;&lt;/div&gt; &lt;/div&gt;   and this is how it appears:    C:\Local\aa.jpg is real file I'm using to try to figure out what the problem is. What am I doing wrong? QGIS ->ArcGIS Server configuration possible? Is it possible to connect a QGIS Desktop 1.8.0 to an ArcGIS Server 10.0 ? and of course publish services on it?.   The configuration would be like this Quantum GIS-> ArcGIS Server 10.0 on Amazon EC2+ POstgreSQL (+PostGIS)-> web app.   Is it possible to create a configuration like this? Do you think I could use it for networking? Viewer-independent copyable spaces at the beginning of a line? What's the proper way to typeset a differential operator? I can't seem to find any consensus on the right way to typeset a differential operator, whether it is:   in a standalone context:  as part of a derivative:  as part of an integral:    In all of these cases, I have seen them sometimes italicized like variables (as in the second and third examples above) and sometimes not italicized like operator names (as in the first example above). There is also additional variation in how much spacing goes between the "d" and whatever it is acting on.  Seeing as this is an incredibly common symbol in many branches of mathematics, I am curious about the lack of standardization -- I've seen all possible combinations used in all possible contexts by many respected authors and publishers.  Are there any rules of thumb that I can follow? I personally tend to prefer the non-italicized version with very little space next to it, so I add:  \renewcommand{\d}[1]{\ensuremath{\operatorname{d}\!{#1}}}   to my default preamble, and use that everywhere, but I'd like to know if I'm breaking any hard-and-fast rules here. Is it okay to use Stack Overflow as the support forum for a product or project? I recently saw a question about a problem (or possible bug) with some product on Stack Overflow. At first, I wanted to suggest that the asker post his question to that product's support forum.  When searching for the URL of that forum, I noticed that the product developer closed its support forums and now advises users to post all their questions on Stack Overflow instead.  Is this within the intended use cases of Stack Overflow, or is this abuse? Kernel of a substitution map No space following macro without argument