How does one solve equations of the form $AX^2+BX+C=0$ where $A,B,C$ are square matrices and $X$ is a matrix to be solved for?  More generally how does one solve equations of the form $AX^2B+CXD+E=0$?  Even more generally how does one solve higher order equations of this form such as $AX^3B+CX^2D+EXF+G=0$  In all cases, I could simply express the entries of the matrix $X$ as variables $x_1,x_2,\dots$ and multiply out the matrices and derive equations corresponding to each entry of the matrices however this would result in (for the first and most simple case at least) a system of 4 quadratic equations in 4 variables which I have no experience with. Is this the best approach or can the fact that we are dealing with matrices help to simplify the problem?  Thanks
Here is the quadratic matrix equation I've been looking at lately:  $$ Q_{r,r}=A_{r,r}X_{r,r}^2+B_{r,r}X_{r,r}+C_{r,r}=0_{r,r} $$  Note that $A, B, C,$ and $X$ are $r \times r$ matrices. $A$ contains known elements, $B$ contains known elements, $C$ contains known elements, and $X$ contains the unknown elements that you are solving for. $ 0_{r,r} $ is just the $r \times r$ null matrix.  Is there any solution for $X$ in terms of $A, B,$ and $C$ (making no easy assumptions)? (e.g. $X$ is a diagonal matrix, $A=B=C$, or anything of that sort.)  I have tried to solve this and nothing has worked out. I attempted solving it generally by manipulating the matrices in variable form (i.e. actually writing out the matrices $A, B, C,$ and $X$ in variables) and finding a unique solution for all of the elements of $X$ in terms of the elements of $A, B,$ and $C$. That didn't work out beyond the case of $r=1$.  Trying to solve it by looking at $r$ at different values did not work out either; I ended up with very abysmal equations at just $r=2$. I don't know exactly how to make this appealing to the denizens of math.stackexchange, but it (as far as I know) isn't a heavily studied problem.  There is a very high possibility that I've just been doing elementary techniques and nothing of note, so I hope someone or a group of people could shed light on this.
Selecting Unique Elements From a List in C# Displaying verbatim content with \tl_to_str:* \documentclass{article} \usepackage{expl3} \ExplSyntaxOn \begin{document} \tl_to_str:n { testing _ things }  \par  \tl_set:Nn \l_tmpa_tl { testing _ things } \tl_to_str:N \l_tmpa_tl \end{document}   gives a strange result (testing ̇things).  How can I get testing_things?  I basically want verbatim content (with line breaking). Show that a matrix $A$ is singular if and only if $0$ is an eigenvalue. I can't find the missing link between singularity and zero eigenvalues as is stated in the following proposition:  A matrix $A$ is singular if and only if $0$ is an eigenvalue.  Could anyone shed some light? How to vertically center a "div" element for all browsers using CSS? If a lens focuses all incoming light to a point, how do we get 2D images? How do lenses produce 2-dimensional images, if a lens bends all incoming rays of light to intersect at the focal point? Shouldn't this produce a single dot of light on a screen placed at the focal length?  This is basically the standard diagram that always shows up in textbooks:    I know this doesn't happen in real life--I used to use telescopes pretty frequently for work. The most in-focus image would be the one with the smallest diameters for the stars, which, of course, we think of point sources at infinity. But--the light from all stars (despite being at infinity) is not all focused at a single point. Instead, each star's light is focused at its own $(x,y)$ point on a 2D image.  I can't reconcile the theory as I understand it with any of my real-life experience with optics. What am I missing? If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$ Noticing that Newtonian gravity and electrostatics are equivalent, is there also a relationship between the general relativity and electrodynamics? In the Upcoming Halloween Update Posted By SOOPUR SELL (supercell), there have been sneak peeks released showing new skeleton traps for TH 8+. I also remember the Santa Strike from one of the Christmas Updates. My question is when these holiday events finish, do we lose the money we spent on the Traps or do we keep them forever? Since Ubuntu 18.04 I can't drag a file e.g. image file to an open application such as Inkscape or  email message window. In Ubuntu 17.10 I was able to select the image file by clicking on it, and while holding down left mouse button I would hit the supper key displaying all open windows where for there I could then drop the file in any open app or folders. This is no longer possible/default behaviour since Ubuntu 18.04. Is there a workaround that would allow me to achieve this or even a way to recreate the same behaviour like in 17.10? Applied boolean modifier leaving shading artifacts with flat shading I tried to manipulate a cube with the Boolean modifier. I tried to cut out an edge of the cube with a cylinder. Also, the shading result looks awful in FLAT SHADING mode, so I turned on SMOOTH SHADING.   Has anyone an idea how to achieve a better result. (I also cut out a Sphere first, this made the result less awful).   I used Blender Internal and rendered with smooth shading for smoothness.  Using flat shading looks okay, except the "round" cutouts :/  Face normals are okay.  First image is smooth shading.    EDIT: User Pisurquatre suggested the EdgeSplit modifier, works perfect to me (Maybe a little tweaking here and there): Why do professional print houses use gamut-limiting CMYK? How to find what device a file is on (and use that in a script)? I want to find out what device my file is on so that I can use it in a script. I can get this far:  $ df  . Filesystem   512-blocks      Used Available Capacity  Mounted on /dev/disk0s2  498438976 294369520 203557456    60%    /   but this output feels too clumsy; is there a better way than parsing this to get the first 'word' of the second line?  What I really need is something like this so I can pipe it to the next command:  $ somecommand . /dev/disk0s2   How can I achieve this, preferably without resorting to string hacking the 'df' output? Eclipse issues warnings when a serialVersionUID is missing.       The serializable class Foo does not declare a static final   serialVersionUID field of type long   What is serialVersionUID and why is it important?  Please show an example where missing serialVersionUID will cause a problem. Connect Tikz nodes via two parallel arrows I am just starting to learn using Tikz and am a little lost within its seemingly infinite amount of options and possibilities.  What I am trying to do is visualize a few Matlab function interaction (which one calls which one, what is the in- and output). Since there are only about six subroutines, I place each one manually as a node. This is an example:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{shapes,positioning}  \begin{document}  \begin{tikzpicture}[auto] \node           [rounded rectangle,draw]        (node1)     {Node 1}; \coordinate [above=of node1]                                (input); \node           [rounded rectangle,draw]    at  (-10:6cm) (node2)       {Node 2}; \draw           [-&gt;,very thick]             ([xshift=0.1cm]  input.south)  --                  node {right}                ([xshift=0.1cm]  node1.north); \draw           [-&gt;,very thick]             ([xshift=-0.1cm] node1.north) --                  node {left}                 ([xshift=-0.1cm] input.south); \draw                                            (node1) -- (node2);  \end{tikzpicture}  \end{document}     Now I would like to apply these double in/out arrows I (probably pretty awkwardly) created using xshift to all node connections, so in this example also for the connection between Node 1 and Node 2. Ideally, the space between these two arrows would be the same, independant of their angle and they would nicely connect to the outer line of the nodes. Any idea how I could achieve this? Similar questions I have found only used rectangle and orthogonal lines. Why directory with large amounts of entries does not shrink in size after entries are removed? Close vote notice: While the linked duplicate asks same question, the comments and the only answer do not provide an answer which draws from authoritative source. The , does in fact provide the answer I've been seeking. Thus the proposed duplicate isn't helpful, however Thomas's answer here is proper.  According to the  and personal account , directories that have large amount of entries get to a size over 4096 bytes (which can be seen with ls -l output), but once the entries are removed - the number never goes down.   The question is "why" ? Is it due to how ext4 filesystem configured to retain directory metadata ? Obviously removing the directory and recreating it isn't a solution, since it deletes original inode and creates a new one. What can be done to decrease the number manually ? Why Didn't People Think That Voldemort Died After His Attack on Baby Harry? After the Avada Kedavra curse rebounded from Harry onto Voldemort, Voldemort disappeared. I'm curious as to why so many key players believed Voldemort was still "alive" somewhere, his powers decimated, but, under the right circumstances, able to return to full power. It seems the witches and wizards in chapter one of Philosopher's Stone, The Boy Who Lived, were happy to believe that Voldemort was gone forever. Why didn't everyone believe Voldemort had died, rather than just disappeared? Was it the lack of a body? Or were there other reasons?  Here's a couple of quotes from Philosopher's Stone if anyone wants to review them. Otherwise, just disregard.     ‘A fine thing it would be if, on the very day You- Know-Who seems to have disappeared at last, the Muggles found out about us all. I suppose he really has gone, Dumbledore?’   ‘It certainly seems so,’ said Dumbledore. ‘We have much to be thankful for.’   Philosopher's Stone - page 13 - Bloomsbury - chapter 1, The Boy Who Lived   ‘But what happened to Vol– sorry – I mean, You-Know-Who?’   ‘Good question, Harry. Disappeared. Vanished. Same night he tried ter kill you. Makes yeh even more famous. That’s the biggest myst’ry, see ... he was gettin’ more an’ more powerful – why’d he go?   ‘Some say he died. Codswallop, in my opinion. Dunno if he had enough human left in him to die. Some say he’s still out there, bidin’ his time, like, but I don’ believe it. People who was on his side came back ter ours. Some of ’em came outta kinda trances. Don’ reckon they could’ve done if he was comin’ back.   ‘Most of us reckon he’s still out there somewhere but lost his powers. Too weak to carry on. ’Cause somethin’ about you finished him, Harry. There was somethin’ goin’ on that night he   hadn’t counted on – I dunno what it was, no one does – but somethin’ about you stumped him, all right.’   Philosopher's Stone - pages 46-47 - Bloomsbury - chapter 4, The Keeper of the Keys How can tangential acceleration from a radial force be explained? A mass is attached to a rope, and put into a circular motion. If I pull the string from the center, the tangential speed of the mass will increase (by conservation of angular momentum).   I am applying a force only in the radial direction, so how can the tangential velocity increase if there is no tangential force? A tool to measure signal strength of wireless I want to know if there exists any command line option or GUI tool (Ubuntu 11.10) so that I can measure the strength of wireless signal at a given location. I have a house where the wireless is not easily accessible at all corners I want to know at which corners or rooms the signal is weak. Vote too old to be changed, unless post is edited Anyone seen this one yet? Vote too old to be changed, unless post is edited My vote was only a few hours old. Then I realized the answer was wrong. Updated my question and went to remove the vote. Even after updating question I still have the issue. Small issue, but frustrating.  See also: Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$