I need to prove that if $A$ is a square invertible matrix then $AA^\top$ ($A$ multiply $A$ transpose) is positive definite.  I tried to prove that all the eigenvalues are positive. I know that $AA^\top$ is symmetric, and its determinant is positive, and trace is positive.   Therefore: product of eigenvalues is positive , and sum of eigenvalues is positive.  In addition $0$ is not an eigenvalue. That's what I know so far but I still cannot conclude that every eigenvalue for itself is positive. What have I missed? Thanks!
Say if $A$ is an $n \times  n$ matrix, why is it that if $A^{T}A$ is positive definite, the matrix $A$ is then invertible? All I know is $A^{T}A$ gives a symmetric matrix but what does $A^{T}A$ is positive definite tell or imply or hint about the matrix $A$ itself that leads to the fact that it will be invertible?
Let $M=\big( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \big)$. Prove $\det(M)=\det(A)\cdot \det(D-C·A^{-1}·B)$. Let  (shown in matrix blocks) $M=\big( \begin{smallmatrix} A &amp; B \\ C &amp; D \end{smallmatrix} \big)$ a square matrix such that $A$ is invertible and $D$ is a square matrix. I have to prove that  $$\det(M)=\det(A)\cdot \det(D-C·A^{-1}·B)$$  I also have an indication to use it: Consider previously these cases: \begin{align*} A_1 &amp;= \begin{pmatrix} A &amp; 0 \\ 0 &amp; I \end{pmatrix},\\ A_2 &amp;= \begin{pmatrix} I &amp; 0 \\ 0 &amp; D \end{pmatrix},\\ A_3 &amp;= A_1·A_2,\\ A_4 &amp;= \begin{pmatrix} I &amp; A^{-1}·B \\ 0 &amp; I \end{pmatrix},\text{ and}\\ A_5 &amp;= A_3\cdot A_4. \end{align*}   The problem is difficult in general for me and I don't even know how to use the indication. What is the best way to give a C# auto-property an initial value? How to roll back Ubuntu to a previous version? I just installed a new version of Ubuntu and I want to roll it back to the previous version.  How can I do that? Is that even possible? Proof of the duality of the dominance order on partitions Could anyone provide me with a nice proof that the  $\leq$ on partitions of an integer $n$ satisfies the following: if $\lambda, \tau$ are 2 partitions of $n$, then $\lambda \leq \tau \Longleftrightarrow \tau &#39; \leq \lambda &#39;$, where $\lambda&#39;$ is the conjugate partition of $\lambda$ (i.e. the transpose of the set of 'dots' which $\lambda$ represents).   I feel like there must be a nice clever and concise/intuitive proof of this, but all I can come up with is an ugly brute force approach based on the definition of sums of the components $\lambda_i$. Could anyone suggest a particularly nice way to obtain this result? Many thanks. C non-blocking keyboard input I'm trying to write a program in C (on Linux) that loops until the user presses a key, but shouldn't require a keypress to continue each loop.  Is there a simple way to do this? I figure I could possibly do it with select() but that seems like a lot of work.  Alternatively, is there a way to catch a ctrl-c keypress to do cleanup before the program closes instead of non-blocking io? This is a follow-up to .     When I click a link to an SE site in the Android app, it gives me an "open with" pop-up that has two options:         Chrome   Stack Exchange         I can then choose to open SE posts like this "Just once" or "Always."      The problem is that when I set the SE app to "always," and I click "open in browser,"   Except, the app crashes when trying to open it in browser.  LogCat (just right after selecting "Open in browser")  01-22 18:03:36.354: E/AndroidRuntime(16064): FATAL EXCEPTION: main 01-22 18:03:36.354: E/AndroidRuntime(16064): Process: com.stackexchange.marvin, PID: 16064 01-22 18:03:36.354: E/AndroidRuntime(16064): java.lang.NullPointerException: Attempt to invoke interface method 'int java.util.List.size()' on a null object reference 01-22 18:03:36.354: E/AndroidRuntime(16064):    at com.stackexchange.marvin.utils.Helpers.getNonMarvinViewIntent(Helpers.java:452) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at com.stackexchange.marvin.ui.question.QuestionDialog$1.onClick(QuestionDialog.java:39) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at com.android.internal.app.AlertController$AlertParams$3.onItemClick(AlertController.java:1108) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at android.widget.AdapterView.performItemClick(AdapterView.java:310) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at android.widget.AbsListView.performItemClick(AbsListView.java:1145) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at android.widget.AbsListView$PerformClick.run(AbsListView.java:3066) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at android.widget.AbsListView$3.run(AbsListView.java:3903) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at android.os.Handler.handleCallback(Handler.java:739) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at android.os.Handler.dispatchMessage(Handler.java:95) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at android.os.Looper.loop(Looper.java:148) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at android.app.ActivityThread.main(ActivityThread.java:5417) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at java.lang.reflect.Method.invoke(Native Method) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:726) 01-22 18:03:36.354: E/AndroidRuntime(16064):    at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:616)   I suspect it's related to , but not sure.  Specs:   Device model: Nexus 5 Android version: Android 6.0.1 SE app version: 1.0.82 (beta) Installed browser: Chrome 47.0.2526.83 (latest at current writing) I am developing a set of scripts, which I want to be present for all users, when I set up a server. But I don't want them in the standard locations like /usr[/local]/[bin],[sbin] etc.  Is there some existing convention for where such scripts should be placed? Why should I "watch" a nearby Pokémon? After update, get "error: file not found." followed by "grub rescue> _" What's the difference between an inverter with a bubble at the input and one with a bubble at the output? See , page 2: the internal logic diagram for the MM74HC138.    The diagram shows logic inverters with bubbles at the input or at the output. Is there an actual difference between them? I thought the way to undo a \def was to define it to \relax, but that does not appear to work.  Also, am not sure why this produces no output if I try to use \def instead of \let in my attempt to do an undef.  \documentclass{article}  \begin{document}  \def\CaseA{}%  \ifdefined\CaseA%     \def\ProcessCaseA{}%     % Do stuff that uses \ProcessCaseA     %\def\ProcessCaseA\relax% no output if use this, and comment next line     \let\ProcessCaseA\relax% no output if comment this, and use above line \fi  % Now do stuff that requires \ProcessCaseA NOT be defined \ifdefined\ProcessCaseA     ERROR: \textbackslash ProcessCaseA is still defined \else     SUCCESS: \textbackslash ProcessCaseA is not defined \fi  \end{document} I'm trying to solve hard combinatorics that involve complicated factorials with large values.  In a simple case such as $8Pr = 336$, find the value of $r$, it is easy to say it equals to this: $$\frac{8!}{(8-r)!} = 336.$$  Then $(8-r)! = 336$ and by inspection, clearly $8-r = 5$ and $r = 3$.  Now this is all and good and I know an inverse function to a factorial doesn't exist as there is for functions like sin, cos and tan etc. but how would you possibly solve an equation that involves very large values compared to the above problem without the tedious guess and checking for right values.  Edit: For e.g. if you wanted to calculate a problem like this (it's simple I know but a good starting out problem) Let's say 10 colored marbles are placed in a row, what is the minimum number of colors needed to guarantee at least $10000$ different patterns? WITHOUT GUESS AND CHECKING  Any method or explanation is appreciated! 8.00GB RAM (3.22GB usable) in Windows 7 x64 I have set up Windows 7 Ultimate 64-bit on my PC. I have 8GB RAM and my BIOS states the correct amount (8192MB), but Windows (System Manager) says I have 8.00GB (3.22GB usable). give me a solution. I'm tried this: Run &gt; MSCONFIG, open the Boot tab &gt; click Advanced Options &gt; make sure that Maximum Memory but this wasn't helpful. My Motherboard Details: CPU Type : DualCore Intel Pentium, 3000 MHz (15 x 200) Motherboard Name : Foxconn G41MXE Motherboard Chipset : Intel Eaglelake G41 System Memory : 3296 MB (DDR3-1333 DDR3 SDRAM) DIMM1: Kingston : 4 GB DDR3-1333 DDR3 DIMM3: Kingston : 4 GB DDR3-1333 DDR3 My BIOS Details: BIOS Type : AMI (03/21/11) Vendor : American Megatrends Inc. Version : 080015 Capabilities : Flash BIOS, Shadow BIOS, Selectable Boot, EDD, BBS System Properties :  Bios File :  What about Bios Update? Editing a post during review doesn't show an "edit not possible" message While reviewing First Posts on Super User, I noticed that I couldn't edit this question (see below). Clicking the &quot;edit&quot; link simply does nothing. Editing is not possible for me right now because I have already five suggested edits pending. It used to be the case that the link would've been grayed out (and the reason shown as a popover). , and while I'm not sure it's an improvement, the current implementation needs some fixing at least.  This bug appears regardless of the actual reason that the edit isn't possible; for instance,  is another user experiencing the same bug who couldn't suggest an edit because of an already pending one. Apparently, I'm far from the only one who experiences this issue. As of last Sunday morning,  have been posted networkwide (and are still alive) and  is #16. Continuity of $\sin(x)/x$ Is $\sin(x)/x$ continuous at $x=0$,(perfectly, not tends to)? The function has right hand and left hand limit same and equals 1 but what happens at perfect zero?it should be undefined. Is it? How to minimize $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + |z_3 - z_4|^2$? If $z_1,z_2,z_3,z_4 \in \mathbb{C}$ satisfy $z_1 + z_2 + z_3 + z_4  =  0$ and $|z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 = 1$, then the least value of $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + |z_3 -  z_4|^2$ is $2$.   I solved this by taking two of the values to be $\frac{1}{2}$ and other two as $-\frac{1}{2}$. But I couldn't prove this in an efficient manner. Do you have to want to hurt someone for the cruciatus curse to work? OK so the scene where Harry sees Voldemort making Draco tutur Rowel brings a few questions to my mind.      he felt the rage that did not belong to him posses his soul, saw a long room, only light by fire light, and the grate blond Death Eater on the floor,screaming and writhing, and a slighter figure standing over him, wand outstretched, while Harry spoke in a high, cold, merciless voice.   "More, Rowel, or shall we end it and feed you to Nagini? Lord Voldemort is not sure he will forgive you this time...You called me back for this, to tell me Harry Potter has escaped again? Draco, give Rowel another taste of our displeasure... do it, or feel, my wrath yourself!"   A log fell in the fire: flames reared, their light darting across a terrified, pointed white face-   I remember in the Order of the Phoenix Harry tries to do the cruciatus curse on Bellatrix, it doesn't work. She say this is because Harry doesn't mean it, he doesn't want to hurt he enough. If she's right about that (I'm sure she is) I don't see how Draco is managing to make to make Rowel writhe and scream, because it is clear he doesn't want to be doing it.   Also Harry was actually angry at Bellatrix and if his anger at her killing his godfather won't work... well you get my drift.   I know Draco is probably better at magic, but I feel Bellatrix is right, you really have to want them to hurt for it to work.  Next thing, Voldemort's having Draco do it why? Is he doing to upset Lucius, because that and lot of effort to go to, seeing as Voldemort is angry and he'd probably do a better job himself. Or is he trying to train himself a new soldier, I mean the other ones are getting old. Are there any known uncountable transfinite increasing sequences of real numbers? The real numbers are uncountable, so assuming the axiom of choice there is at least transfinite sequence of real numbers $r_0, r_1, r_2, ..., r_\omega, r_{\omega + 1}, ..., $ up to (and possibly including) $r_{\omega_1}$ that contains an uncountable number of terms. However, it's not guaranteed that $r_0 &lt; r_1 &lt; ..., r_\omega &lt; ...$  Is there a known transfinite sequence of real numbers that is monotonically increasing and which contains uncountably many real numbers?  Thanks! Message appearing when a user was removed.     It seems I lost 1765 reputation points on Stack Overflow yesterday. The reason given in my history is "removed", with the description of "User was removed ()". I don't understand what that means. What caused this? When and why will a compiler initialise memory to 0xCD, 0xDD, etc. on malloc/free/new/delete?