Let $\lambda_{\min}$ be the smallest eigenvalue of the positive definite matrix $\mathbf{S}$, and $\|\mathbf{a}\|=r$. Then  $$ \mathbf{a}^T \mathbf{S}\mathbf{a} &gt; \frac{1}{2}\lambda_{\min} r^2  $$  What properties of matrices were used to obtain the result? Thanks!
I got a positive definite matrix $B,$ that is, $V(x) = x^TBx &gt; 0$ for any vector $ x \neq 0.$  I want to show that $ \lambda_\min \|x\|_2^2 \leq V(x) \leq \lambda_\max \|x\|_2^2$ for any $x \neq 0,$ where $\lambda_\min$ and $ \lambda_\max$ are defined by $$ \lambda_\min = \min \lbrace | \lambda|: \lambda \text{ is an eigenvalue of } B \rbrace$$ and  $$ \lambda_\max = \max \lbrace | \lambda|: \lambda \text{ is an eigenvalue of } B  \rbrace$$  Any hint please?
Execute shell commands in Python How to extrude and scale with an even offset? I'm quite new to Blender, so forgive me if this is a dumb question. I don't even know hot to describe it properly, so:      Basically I want to extrude a rectangle and then scale it along two axis, but without keeping the proportions; I want to reduce them of the same quantity, so that I get something similar to the frame of a picture. Remove Meta Privilege for MSE (only) Was the Wookiee language fully designed or is it just random emotionally-relevant sounds? We know that some of the other Star Wars languages were fairly well developed by the movie's creators, in terms of vocabulary, grammar, etc.  Was the Wookiee language designed in this level of detail or did they just try to put together sounds that made sense emotionally? What is the remainder when $2^{1990}$ is divided by $1990$? I actually do not have the basic idea on how to approach these type of questions....so please tell me a generalized method about all this too.  It came in RMO, and the question is:  What is the remainder when $2^{1990}$ is divided by $1990$ ? Problems with fonts I'm an average user of LaTeX, but not proficiency. I'm trying to learn from book templates I've found on the web, but I get this error message:  Trying to make PK font ccicons at 1493 DPI... Running miktex-makemf.exe... miktex-makemf: The ccicons source file could not be found. Running ttf2pk.exe... miktex-makemf: PK font ccicons could not be created.  LaTeX Font Warning: Some font shapes were not available, defaults substituted.  !pdfTeX error: pdflatex.exe (file ccicons): Font ccicons at 1493 not found  ==&gt; Fatal error occurred, no output PDF file produced!   I'm using MiKTeX (over Windows 7), and according to the MiKTeX package manager the "ccicons" package is already installed.  Could you please help me solve this problem? How to remove desktop environments? I installed few environments that I wanted to try out on Ubuntu 12.04, but none of them worked at all. It could be that I installed them all at the same time, meaning the OS didn't get a chance to work everything out, but either way, they didn't work. I would now like to remove them.   The one's that I installed are Xubuntu/Xfce, Lubuntu/LXDE, Kubuntu/KDE, Fluxbox, OpenBox, cinnamon, and MATE. I installed them. Everything seemed to have been working properly until I actually tried to use the shells and nothing loaded at all. Except for Fluxbox, I think that one worked. I want to know of any way to repair or perhaps just remove the packages entirely. I might have already removed them because I did the apt-get remove command on all of them, but they were still in the list on the login screen. regular expressions / pattern matching on ubuntu command line Can you cast a spell at a higher level, even if it doesn't have an "At higher levels" entry? Primes dividing a polynomial Let $g(x)\in \mathbb{Z}[x]$, a nonconstant polynomial. Show that the set of primes $p$ such that $p\mid g(n)$ for some $n\in \mathbb{Z}$ is infinite.  I don't know how to start. I have tried asuming that the set is finite but i haven't reached any contradiction. Eigenvalues of a matrix whose square equals its transpose Let $A$ be a $n\times n$ matrix with $A^2=A^t$. Show that every real eigen value of $A$ is either $ 0$ or $1$ Compilers and argument order of evaluation in C++ Okay, I'm aware that the standard dictates that a C++ implementation may choose in which order arguments of a function are evaluated, but are there any implementations that actually 'take advantage' of this in a scenario where it would actually affect the program?  Classic Example:  int i = 0; foo(i++, i++);   Note: I'm not looking for someone to tell me that the order of evaluation can't be relied on, I'm well aware of that. I'm only interested in whether any compilers actually do evaluate out of a left-to-right order because my guess would be that if they did lots of poorly written code would break (rightly so, but they would still probably complain). I was wondering if it is possible to get a link to a rigorous proof that  $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$ How to prevent form from being submitted? Links in newsletter preview link to YouTube The problem: The links at the bottom of the Stack&nbsp;Exchange newsletter link to a YouTube video instead of the right place on the website. I´m not sure if this is an oversight or a clever hack.  Screenshot: Is there a difference between Bayesian and Classical sufficiency? The title pretty much says it all. I wonder whether there is any difference in the way Bayesians understand sufficiency vs. the way orthodox statistics understands sufficiency, or are they equivalent?  If there is a difference, what is it? Is a switches voltage and current rating a hard limit? Should the first word after a colon be capitalized? Should the first word after a colon be capitalized? Which of the following is correct?   For example: This. For example: this. How does a transformer transmit power from the primary to the secondary? I'm trying to get my head wrapped around transformer operation and in the process regretting the times I snoozed in my Electromagnetics class as a EE student back when I was a lad :)  I'm looking for an intuitive understanding, but not just an analogous one. I'd like it to be grounded in the actual physics of what's happening. I've found several excellent sources on the web, but they all seem to skirt this question.  I've come across a few interesting hints and am now tantalizingly close, I think, but still yearning :)  Fact 1: Although varying sinusoidally, the "peak-to-peak" flux, so to speak, in a transformer's core is essentially constant (for a given voltage applied to the primary), regardless of the load.  My intuitive hypothesis was that variation in the "strength" of the flux was what transferred the energy from the primary to the secondary, but this fact would seem to contradict that theory. I had thought that the primary makes a bunch of flux based on the current flowing through it and the secondary sucks it up to make current of its own. No dice, it seems.  Then of course there's the fact that the formula for flux involves only voltage, time (frequency), and turns :)  Fact 2: The current in the primary is (approximately) 90 degrees out of phase with the voltage at no load, and approximately aligned in phase at full load.  This fact seems very promising and also curiously satisfying. It would imply that the Volt-Amps (VA) of the primary is constant and only the power factor changes as the current load on the secondary increases.  But I still don't get how the energy is actually being transferred. It seems vaguely like the flux is just there as an energy conductor or something and some other phenomenon is actually doing the energy transfer bit.  Can someone see what I'm missing and explain what's actually happening in there? Beamer consists of multiple frame layouts