Let $A\in M_{n,n}(\mathbb{C})$ be a diagonalisable matrix.  Prove $\exists B \in M_{n,n}(\mathbb{C})$ such that $B^{2016} = A$  I can't see why this statement would be true. Perhaps I'm missing something.  Can you assist me please?
Is there a general method (which can be implemented by hand) to finding the $n$-th roots of $2 \times 2$ matrices? Is there a similar method for a general $m \times m$ matrix? (for $n &gt; 1$ and $n\in\mathbb{Z}$)
Asymptotics of uniformly continuous functions over $\mathbb R$ How to mesh codeigniter with ajax $A=\begin{pmatrix}0&amp;0&amp;1\\ 0&amp;1&amp;0\\ 1&amp;0&amp;0\end{pmatrix}$  How do I show that there are no solutions for $X^2=A$ in $M_3(R)$?  I tried $X^2=\begin{pmatrix}a_{11}&amp;a_{12}&amp;a_{13}\\ a_{21}&amp;a_{22}&amp;a_{23}\\ a_{31}&amp;a_{32}&amp;a_{33}\end{pmatrix}^2=A$, but I couldn't seem to find anything that shows there are no solutions. How should I approach this? It looks like we'll be adding  support to Stack Overflow. This is necessary to prevent bots, spammers, and other malicious scripted activity. We only want human beings to post or edit things here!  We'll be using a JavaScript (jQuery) CAPTCHA as a first line of defense:    The advantage of this approach is that, for most people, the CAPTCHA won't ever be visible!  However, for people with JavaScript disabled, we still need a fallback and this is where it gets tricky.  I have written a  which we can re-use.    However, I'd prefer to go with something textual to avoid the overhead of creating all these images on the server with each request.  I've seen things like..   ASCII text captcha: \/\/(_)\/\/ math puzzles: what is 7 minus 3 times 2? trivia questions: what tastes better, a toad or a popsicle?   Maybe I'm just tilting at windmills here, but I'd like to have a less resource intensive, non-image based &lt;noscript&gt; compatible CAPTCHA if possible.  Ideas? I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, there must exist a linear functional from $\ell_2$ into $\mathbb{R}$. However, I'm having trouble actually coming up with it.   I believe I'm supposed to find an unbounded function (although I'm not sure why an unbounded function is necessarily not continuous; some light in that regard would be appreciated too), so I thought of using the vectors $e^i$, which have all entries equal to zero, except for the $i$-th one. Then, you can define $f(e^i)=i$.  That'd be unbounded, but I'm not sure if it'd be linear, and even if it is, I'm not sure how to define it for all the other vectors in $\ell_2$.   A friend mentioned that at some point the question of whether the set $E=\{e^i:i\in\mathbb{Z}^+\}$ is a basis would come up, but I'm not sure what a basis has to do with continuity of $f$.   I'm just learning this topic for the first time, so bear with me please.   The space of sequences that are eventually zero (suggested by a few people) turned out to be exactly what I needed.  It also helped to cement the notions of Hamel basis, not continuous, etc. How to remove selected commit log entries from a Git repository while keeping their changes? I would like to remove selected commit log entries from a linear commit tree, so that the entries do not show in the commit log.  My commit tree looks something like:  R--A--B--C--D--E--HEAD   I would like to remove the B and C entries so that they do not show in the commit log, but changes from A to D should be preserved. Maybe by introducing a single commit, so that B and C become BC and the tree looks like.  R--A--BC--D--E--HEAD   Or, ideally, after A comes D directly. D' representing changes from A to B, B to C and C to D.  R--A--D'--E--HEAD   Is this possible? if yes, how?  This is a fairly new project so has no branches as of now, hence no merges as well. Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$ Why is ++i is l-value and i++ not? /proc/PID/fd/X link number How do you reset timers? How to get class object's name as a string in Javascript? If $X$ is an orthogonal matrix, why does $X^TX = I$? It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of $X$ are orthonormal.   Can you explain this better to me? I tried the $2\times 2$ case, but it isn't clear to me why this is true. Why do I sometimes see a downloading 'pie' icon when viewing photostream photos? it normally shows in the bottom corner as a sort of 'pie' that fills up as the picture is downloaded, similar to the 'app downloading' pie when viewing a downloading app on the home page/s  prior to this the picture resolution is terrible Health effect of electromagnetic and acoustic waves on brain function How to concatenate items in a list to a single string? I use TeX Live 2010 and my editor is Texmaker Bidi. It was working probably good but from 2 months ago I've encountered with the error:   !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! Fatal fontspec error: "cannot-use-pdftex" !  ! The fontspec package requires either XeTeX or LuaTeX to function. !  ! You must change your typesetting engine to, e.g., "xelatex" or "lualatex" ! instead of plain "latex" or "pdflatex". !  ! See the fontspec documentation for further information. !  ! For immediate help type H &lt;return&gt;. !...............................................   when I'm trying to load xepersian LaTeX.  but I do not know how can I change pdfLaTeX to XeLaTeX.    This is the second edit of my question .   the code is :    ‎\documentclass{article}‎ ‎\usepackage{pstricks}‎ ‎\usepackage{‎pst‎-‎bezier}‎ ‎‎\usepackage{xepersian}‎  ‎\settextfont{XB Zar}‎  ‎\setlatintextfont{Junicode}‎  ‎\setdigitfont{XB Zar}‎  ‎\begin{document}‎  ‎!‎سلام دنیا‎‎ ‎\begin{pspicture}(‎3‎,3)‎ ‎\psbcurve(1,2)(5,7)(2,2)(12,3)‎ ‎\end{pspicture}‎ ‎ ‎\end{document}   but the output does not include the shapes . Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)&lt;3$   for all positive integers $n$   This problem is copied from Math Olympiad Treasures by Titu Andreescu and Bogdan Enescu.They start by stating that induction wouldn't directly work here since the right hand side stays constant while the left increases.They get rid of this problem by strengthening the hypothesis.$$\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)\le3-\dfrac{1}{n}$$ and then proceed by induction.  The problem is that I can't find a motivation for the above change.I mean,we could have subtracted a lot of things from the RHS but what should nudge us to try $\dfrac{1}{n}$?The rest of the proof is quite standard,but I can't see how I am supposed to have thought of it.Is it just experience?Or is it a standard technique?A little guidance and motivation will be appreciated. When you cast invisibility on your turn, and walk away from 2 enemies, do they get an attack of opportunity? When a PC is in melee with a creature, casts the basic invisibility spell, and leaves melee, does an NPC get an attack of opportunity? I have two questions. First, which command lists files and directories, but lists directories first?  Second question: I want to copy a list of files into a single directory, but make the target directory the first filename in the command. What's the deal with a leading underscore in PHP class methods? While looking over various PHP libraries I've noticed that a lot of people choose to prefix some class methods with a single underscore, such as  public function _foo()   ...instead of...  public function foo()   I realize that ultimately this comes down to personal preference, but I was wondering if anyone had some insight into where this habit comes from.  My thought is that it's probably being carried over from PHP 4, before class methods could be marked as protected or private, as a way of implying "do not call this method from outside the class".  However, it also occurred to me that maybe it originates somewhere (a language) I'm not familiar with or that there may be good reasoning behind it that I would benefit from knowing.  Any thoughts, insights and/or opinions would be appreciated.