Definitions of positive definiteness usually look like this:     A symmetric matrix $M$ is positive definite if $x^T M x &gt; 0$ for all vectors $x \neq 0$.   Why must $M$ be symmetric? The definition seems to make sense for general square matrices.
Do positive semidefinite matrices have to be symmetric? Can you have a non-symmetric matrix that is positive definite? I can't seem to figure out why you wouldn't be able to have such a matrix, but all my notes specify positive definite matrices as "symmetric $n \times n$ matrices."  Can anyone help me with an example of a non-symmetric positive definite matrix, or some insight into a proof for why it would need to be symmetric should that be the case? Thanks!
"I use to", or "I used to" Which is the correct sentence, if there is a correct one?     I use to be a hitman.        I used to be a hitman.   I've read the second sentence recently in a book, but I was sure it should be I use to be a hitman. How to secure database passwords in PHP? How can I find out if Ubuntu supports a particular hardware component, that is, if it works in Ubuntu?  I am going to put together a computer and I am choosing components.   How do I know if Ubuntu supports this or that motherboard, or this or that sound card, or this or that graphics card, etc? How do I know that there is no conflict in Ubuntu between this and that hardware component?   What resources are there to help with such situations?  I know that there are computers with Ubuntu pre-installed and that there are  for whole computers but this is not what I'm looking for. I'm looking for resources to find out support for particular hardware components and conflicts among particular hardware components. Show that any set in a metric space can be written as the intersection of open sets Show that any set contained in the metric space $(X, d)$ can be   written as the intersection of open sets.   Definitions: A set $A \subseteq X$ is open if $\forall x \in A$, $\exists \varepsilon&gt;0$ such that $B_{\varepsilon}(x) \subseteq A$. A set $C \subseteq X$ is closed if and only if $X \setminus C$ is open.   I also know that the intersection of a finite collection of open sets is open and that any open ball contained in $X$ is open. How can I prove this question? What good CAD programs are available? How do I find out which dlls an executable will load? If I have a Windows executable, how can I find out which dlls it will load?  I'm just talking about which ones that will be loaded statically, not ones it might load dynamically with something like LoadLibrary. Using same modulus for RSA TexShop will not compile without trashing aux files after every error I have a couple of LaTeX files with the same problem. If I produce an error, fix it, and then try to compile a pdf, I get the following error.  )Runaway argument? {{ ! File ended while scanning use of \@newl@ablel. &lt;inserted text&gt;                 \par l.90 \begin{document}   If I then click on the console or trash my .aux file, it compiles fine. Any thoughts on what is going on and how to prevent this annoying extra step? Controlling the value of 'this' in a jQuery event I have created a 'control' using jQuery and used jQuery.extend to assist in making it as OO as possible.  During the initialisation of my control I wire up various click events like so  jQuery('#available input',              this.controlDiv).bind('click', this, this.availableCategoryClick);   Notice that I am pasing 'this' as the data argument in the bind method. I do this so that I can get at data attached to the control instance rather from the element that fires the click event.   This works perfectly, however i suspect there is a better way  Having used Prototype in the past, I remember a bind syntax that allowed you to control what the value of 'this' was in the event.  What is the jQuery way? Is there an easy way to 'break' the axis in a pgfplots environment? What I'm talking about is something like     Any minimal plot will do as an example...for example  \documentclass{minimal}   \usepackage{tikz} \usepackage{pgfplots}  \begin{document} \begin{tikzpicture} \begin{axis}[ymin=0, ymax=80] \addplot {x*0};  \addplot {x^2+50}; \end{axis} \end{tikzpicture}  \end{document}   For example, in the below plot,    I want to break the axis from around y = 10 to y = 40.   Any ideas? Does PGF have this functionality? A search for `break' in the manual doesn't yield anything. I have a raster image (Sentinel-2 band 4) and a shapefile covering the same area than the image (same projection and extension). The shapefile has different polygons which are agricultural fields.   I want to create a new file where, for each polygon, the mean value of the pixels that are inside it is calculated.  Could you please help me to find an appropriate R code for that? Is there a definition of  and proper explanation of it?  It is said "Electric charge is the physical property of matter that causes it to experience a force when close to other electrically charged matter."  How is it though that matter can get charged?  Defining charge as the property of feeling a force with other charged matter seems circular.  What is charge?  Is there a non-circular definition / explanation? Get bytes from std::string in C++ I'm working in a C++ unmanaged project.  I need to know how can I take a string like this "some data to encrypt" and get a byte[] array which I'm gonna use as the source for Encrypt.  In C# I do    for (int i = 0; i &lt; text.Length; i++)     buffer[i] = (byte)text[i];   What I need to know is how to do the same but using unmanaged C++.  Thanks! Is it possible to install MySQL 5.5 or 5.6 on Ubuntu 16.04? I recently upgraded to Ubuntu 16.04. Unfortunately I need to work MySQL 5.5 or 5.6. I tried many solutions to have MySQL 5.5 or 5.6 run on Ubuntu 16.04 like this  . But none of them is working. Ubuntu detect the unmet dependencies but says "but it is not going to be installed" for 5.5 or says Package 'mysql-server-5.6' has no installation candidate...  Is it not possible to force an earlier version of MySQL on Ubuntu? Why is that? Recommendations for Stock Trading and Investment Applications? New "[tag:" syntax interferes with Markdown links Links in [link text][number] format don't work properly if they follow a tag using the new [tag:tagName] format. Here's the result of [tag:python] [gold badge page][1]:      [gold badge page]   Putting the tag inside the link is worse. Here's the result of [[meta-tag:bug] some text][2], where [2] is google.com:        The part of the link outside the tag just disappears silently.  EDIT: To be more precise, the entire link to [2] disappears. The markup that is produced contains (invalid) nested anchor tags, as shown here from Firefox's "View Source":  &lt;a href="http://google.com" rel="nofollow"&gt;&lt;a href="/questions/tagged/bug" class="post-tag required-tag" title="show questions tagged 'bug'" rel="tag"&gt;bug&lt;/a&gt; some text&lt;/a&gt;   So the tag is still a link, but it is a link to the tag page, which is the behavior it would have if the link to [2] didn't exist at all. The user's desired behavior in this case is... ambiguous, to be charitable.  EDIT 2: I briefly included markup from Firebug here, but it turned out to be irrelevant. It has now been removed to improve readability. See  if you're interested. Remove only the commas present within the double quotes In a text file, I want to remove , (commas) and also the " (quotes) (only if the double quotes  contains  numbers separated by commas).  56,72,"12,34,54",x,y,"foo,a,b,bar"   Expected ouput  56,72,123454,x,y,"foo,a,b,bar"   Note: I show the above line just as an example. My text file contains many lines like above and the numbers separated by commas present within the double quotes should vary. That is,  56,72,"12,34,54",x,y,"foo,a,b,bar" 56,92,"12,34",x,y,"foo,a,b,bar" 56,72,"12,34,54,78,76,54,67",x,y,"foo,a,b,bar" 56,72,x,y,"foo,a,b,bar","12,34,54" 56,72,x,y,"foo,a,b,bar","12,34,54","45,57,84,92","bar,foo"   Expected output:  56,72,123454,x,y,"foo,a,b,bar" 56,92,1234,x,y,"foo,a,b,bar" 56,72,12345478765467,x,y,"foo,a,b,bar" 56,72,x,y,"foo,a,b,bar",123454 56,72,x,y,"foo,a,b,bar",123454,45578492,"bar,foo"   There a n number of numbers present within the double quotes separated by commas. And also leave the double quotes which contains characters as it is.  I love sed text processing tool. I'm happy if you post any sed solution for this. Evaluate $ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots$ Evaluate $$ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots$$  All i could do was to see that $$\frac{1}{3}=\frac{1}{2.1+1},\frac{1}{5}=\frac{1}{2.2+1},\frac{1}{7}=\frac{1}{2.3+1},\dots$$  SO, we should be able to write   $$ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots=1+\sum_{n=1}^{\infty}\frac{1}{2n+1}.\frac{1}{4^n}$$  This is what i could simplify the given question to...   But, this is doing me no good.. I could not go any further..  I would be thankful if some one can help me to clear this.  I would request users who are trying to help me to just give necessary hints but not post it as answer.  Thank You. Union of a finite set and a countably infinite set is countably infinite Chess Piece Combinations I came across a question yesterday about combinations, and I wanted to know what the correct answer was. The question states as follows:  There are 8 spaces that are alternately black and white. There is one king, one queen, two identical rooks, two identical bishops, and two identical knights. The king needs to be surrounded by the two rooks. Then, the two bishops can be put on any of the remaining spaces as long as one is on a black space and the other is on a white space. Finally, the queen and the knights can be placed anywhere in the remaining spaces in relation to the other pieces. How many possible arrangements are there considering these conditions?