I've often seen this property used when dealing with covariance matrices, which have just have every desirable property imaginable (e.g. they are symmetric, contain only real numbers, and are positive semi-definite), but I don't recall every seeing any use of this property outside of this context. Is it just a useful property of covariance matrices? Or is it a result of some of their non-unique properties, and if so, which properties?
I have a very simple question that can be stated without proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand Principal components and it is cruucial for me to see the basis of eigenvectors.
It is a theorem in elementary number theory that if $p$ is a prime and congruent to 1 mod 4, then it is the sum of two squares. Apparently there is a trick involving arithmetic in the gaussian integers that lets you prove this quickly. Can anyone explain it? Not sure if someone else has already raised this - I've got eleven notifications that I've received a Caucus badge on English Language Learners. Perhaps one notification for each community that I'm a member of?  I am a member of the community, and did visit the election. Also, the badge hasn't been awarded more that once (so I think it's just the notification that's at fault).  Note that this seems to happen for other badges, it has been reported for: Caucus, Nice Answer, Enthusiast. The advantages and disadvantages of using Kaiser Rule to select the number of principal components I am a student who has taken the basic calculus courses but I am working through Calculus (the fourth edition) by Spivak in my spare time in order to both review the material and to gain a more rigorous understanding of the concepts. I am currently in Chapter 8 where he supplies a proof of the basis of the Intermediate Value Theorem, Theorem 7-1 (pp. 135-136 of the fourth edition). It seems mostly straightforward but I think there is some nuance of it that I don't grasp.  I am including in this post the statement Theorem 6-3 since he references it in the proof of Theorem 7-1 as well as part of Theorem 7-1 itself.     Theorem 6-3 Suppose $f$ is continuous at $a$, and $f(a) &gt; 0$. Then $f(x) &gt; 0$ for all $x$ in some interval containing $a$; more precisely, there is a number $\delta &gt; 0$ such that $f(x) &gt; 0$ for all $x$ satisfying $|x-a| &lt; \delta$. Similarly, if $f(a) &lt; 0$, then there is a number $\delta &gt; 0$ such that $f(x) &lt; 0$ for all $x$ satisfying $|x-a| &lt; \delta$.     Problem 6-16 is also referenced. However,it is the same thing as Theorem 6-3 for one-sided limits. The following proof is the Theorem 7-1 up through the paragraph which I don't fully understand.     Theorem 7-1: If $f$ is continuous on $[a,b]$ and $f(a) &lt; 0 &lt; f(b)$, then there is some number $z$ in $[a,b]$ such that $f(x) = 0$.  Proof: Define the set $A$ as follows:  $$A = \{x : a \le x\le b, \mbox{ and } f \mbox{ is negative on the interval } [a,x] \}.$$  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Clearly $A \ne \emptyset$, since $a$ is in $A$; in fact, there is some $\delta &gt; 0$ such that $A$ contains all points $x$ satisfying $a \le x &lt; a + \delta$; this follows from Problem 6-16, since $f$ is continuous on $[a,b]$ and $f(a)&lt;0$. Similarly, $b$ is an upper bound for $A$ and, in fact, there is a $\delta &gt; 0$ such that all points $x$ satisfying $b-\delta &lt; x \le b$ are upper bounds for $A$; this also follows from Problem 6-16, since $f(b) &gt; 0$. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; From these remarks it follows that $A$ has a least upper bound $\alpha$ and that $a &lt; \alpha &lt; b$. We now wish to show that $f(\alpha) = 0$, by eliminating the possibilities $f(\alpha) &lt; 0$ and $f(\alpha) &gt; 0$. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Suppose first that $f(\alpha) &lt; 0$. By Theorem 6-3, there is a $\delta &gt; 0$ such that $f(x) &lt; 0$ for $\alpha - \delta &lt; x &lt; \alpha + \delta$. Now there is some number $x_0$ in $A$ which satisfies $\alpha - \delta &lt; x_0 &lt; \alpha$ (because otherwise $\alpha$ would not be the least upper bound of $A$). This means that $f$ is negative on the whole interval $[a,x_0]$. But if $x_1$ is a number between $\alpha$ and $\alpha+\delta$, then $f$ is also negative on the whole interval $[x_0,x_1]$. Therefore $f$ is negative on the interval $[a,x_1]$, so $x_1$ is in $A$. But this contradicts the fact that $\alpha$ is an upper bound for $A$; our original assumption that $f(\alpha) &lt; 0$ must be false.    The point I don't understand is in the final paragraph of the excerpt. Why does Spivak split the interval $[a,b]$ up into the subintervals $\lbrack a,x_0 \rbrack$ and $[x_0,x_1]$? Isn't it true that, because $\alpha$ is a least upper bound of $A$ and $f$ is continuous on $[a,b]$, it must be true that if $a \le x &lt; \alpha$ then $f(x)&lt;0$? I understand the need to pick out the $x_1$ but not the $x_0$. Especially perplexing is the statement "there is some number $x_0$ in $A$ which satisfies $\alpha - \delta &lt; x_0 &lt; \alpha$ (because otherwise $\alpha$ would not be the least upper bound of $A$." Wouldn't it be true that if there were some $x_0$ such that $a \le x_0 &lt; \alpha$ and $f(x_0) \ge 0$ that $x_0$ would be an upper bound for $A$, contradicting that $\alpha$ is the least upper bound of $A$?  Thanks for any help. Is it possible to add an email just below the authors in this code?  \documentclass[11pt,oneside,a4paper]{article} \usepackage{hyperref}   \title{Hello} \author{User1 \and User2}    \begin{document} \maketitle  \section{Introduction} A1 \section{Literature Review} A2  \section{Conclusion} A2  \end{document} USB Charge and data separate cables Is it possible to split micro USB to USB cable so two cables (data + and data -) go to the PC with USB and two other power cables go to power brick? How to design partitions for SSHD? I'm going to install Linux on Lenovo B5400, which contains:   500GB Hybrid SSHD with a 8 GB SSD and 500 GB HDD   Designing partitions for my 2 Linux distros was quite easy before, when I had only HDD. I always had:   10GB for Linux distro (mounted as "/") 10GB for another distro (again "/"; for another distro; for testing purposes or to assure continuity when switching distros) 2GB for swap (used for both distros; no need to separate) 300GB+ for "/home" (used for both distros; different usernames eliminates possible conflicts)   It's simple, isn't it? I would like to keep this approach, but I know that SSD has its own specificity - it prefers to be read, continuous writing may affect its life span. I don't want to kill SSD in a year, so I would like to move some system partitions to HDD in order to protect SSD. As far as I know /tmp is one of often read&amp;written locations. How much space it needs? What else should be placed on HDD?  Or: how to design partitions to achieve exactly the same effect and to keep SSD as safe as possible?  I'd be grateful for your help. Multiple "order by" in LINQ I have two tables, movies and categories, and I get an ordered list by categoryID first and then by Name.  The movie table has three columns ID, Name and CategoryID. The category table has two columns ID and Name.  I tried something like the following, but it didn't work.  var movies = _db.Movies.OrderBy( m =&gt; { m.CategoryID, m.Name }) If there is one noble left in line and you play "fled to England" to discard it (thus, ending the day) do you collect a noble from the new day? Viewing all my questions from Android app How can I detect inside a TeX file which engine (pdftex/xetex/luatex) is used to compile the file? Is there a LaTeX package that provides this functionality? The choice field adds a blank option in all my sharepoint lists. How do I remove this? the column settings does not have a blank option. Can an infectious diseases come from a plant? Coronavirus, HIV, 1918 Flu, etc. They all come from animals. Do any infectious diseases (in humans) come from plants? More specifically, are there viruses that infect plants that can mutate to infect humans? How can I show notify-send messages triggered by crontab? Difficulty in creating macro '\newcommand' for the beginning and end of frames in Beamer I would like to create a Latex macro for my beamer presentations. I use beamer frequently, and want to make macros for the \begin{frame} and \end{frame} to require fewer keystrokes. At the beginning of the document I have:  \newcommand{\bfrm}{\begin{frame}}%start new frame \newcommand{\efrm}{\end{frame}}%end new frame   Which compiles fine. I then test the first of the two macros for the beginning of the frame:  \bfrm{New frame testing}     blah blah \end{frame}   Which produces the result I am looking for. I then test the second of the macros:  \bfrm{New frame testing}     blah blah \efrm   Fails to compile, saying that there is a runaway argument. What is the reason for the failure? What are the complexity guarantees of the standard containers? We often get questions on bicycles.stackexchange about identifying a bicycles manufacturer, model, make, and year.  Often these questions are downvoted or closed -- usually with a comment attached to them that it's not necessary to know the exact model/make/year of a bicycle if you want to fix, repair, or ride your bike. Why should owners not care about the model/make/year of their bicycle? Especially when it comes to less expensive bikes (aka BSOs) and BMX bicycles? This is intended as a canonical question that we can point closed questions to. Also see      (generally not helpful for identifying a brand/model) Create UEFI bootable USB stick to install Windows AND Ubuntu from it I know how to make a UEFI bootable stick to install Ubuntu (explained ) and I know how to make one to install Windows 7 (, sorry). I guess for Win 8 the procedure would be similar, but I haven't looked into it.  I also know how to make a legacy bootable stick from which I can install Windows and Ubuntu using YUMI. I have failed so far though at creating a UEFI bootable stick that can do this. When I do a UEFI boot from my YUMI multiboot stick it gives me no option to choose between Ubuntu and Windows (hardly surprising I guess) and instead simply UEFI boots Windows (which I find kind of surprising to be honest but maybe this is due to the fact that I added the Windows image after the Ubuntu image and this simply replaced Ubuntu's EFI boot files).  So how would I proceed to create such a bootable stick? Can an Indian citizen visit Dubai on a layover? Search and replace multiple values with multiple/different values in PHP5?