On page 9 of Linear Regression Analysis 2nd Edition of Seber and Lee there is a proof for the expected value of a quadratic form that I don't understand.  Let  $X = (X_i)$ be an $n \times 1 $ random vector and let $A$ be an $ n \times n$ symmetric matrix. If $\mathbb{E}[X] = \mu$ and $\operatorname{Var}[X] = \Sigma = (\sigma_{ij})$ then $\mathbb{E}[X^T AX] = \operatorname{tr}(A\Sigma) + \mu^T A\mu$  The problem I have is almost right out the gate, I can't see how $\mathbb{E}[X^T AX] = \operatorname{tr}(\mathbb{E}[X^T AX])$ I think I get the rest of the proof, but if someone here can't point me in the right direction on this part, I'd be forever grateful!
For random variables $X \in \mathbb{R}^h$, and a positive semi-definite matrix $A$: Is there a simplified expression for the expected value, $\mathop {\mathbb E}[Tr(X^TAX)]$ and variance, $Var[Tr(X^TAX)]$? Please note that $A$ is not a random variable.
Physically what does warping (of space-time) mean? So there's general relativity and Einstein's field equations that tell us "mass(or equivalently energy) warps space-time, and the warping tells mass how to move", but I'm still having trouble understanding how space (a thing I conceive to be 'empty' or nothingness) can be 'warped'.  After researching books, on-line articles, and this stack exchange itself I can't seem to find any descriptions or discussions that don't immediately resort to jumping into GR and its mathematics to provide a more physical intuitive understanding of what it means to for space to be warped.  Is it space, this empty thing, that's getting warped. Or rather is it some field (Higgs field?) that exists around matter that's getting warped?  Although this question appears similar to  , it's asking if it's empty space that's being warped by matter or rather a field in space that's being warped (the Higgs Field?). The other question and all 10 answers addressing it do not address fields vs. empty space - What's really getting warped? installing rtl8814AU wifi adapter driver on ubuntu 16.04 Every Christmas, Harry receives a mediocre gift from the Dursleys. In Harry Potter and the Chamber of Secrets he gets a toothpick.     At that moment, Hedwig swooped into the room, carrying a very small package in her beak.     “Hello,” said Harry happily as she landed on his bed. “Are you speaking to me again?”     She nibbled his ear in an affectionate sort of way, which was a far better present than the one that    she had brought him, which turned out to be from the Dursleys. They had sent Harry a toothpick    and a note telling him to find out whether he’d be able to stay at Hogwarts for the summer    vacation, too.    They hate owls, but obviously use them to contact Harry at school? Are human chromosomes connected or separate molecules? Do the 46 human chromosomes form a single unbroken DNA helix?  Or is it rather that a human's genome consists of 46 disconnected helices?    If it is the former, does the common numbering scheme for the chromosomes have any correlation to their actual ordering in the one large strand?  If is the latter, is there a convention on how the chromosomes are ordered in genomic datasets?  Also, is there a clear understanding of how sister chromosomes "find" each other in Meiosis I?  Generally, during periods when Mitosis/Meiosis are not occurring,  what's a good physical picture for how the chromosomes are physically arranged (e.g. a bowl of 46 spaghetti noodles, or maybe the sister chromosomes always stay close together, etc)  thanks! Suppose I have a program named any_program.exe and my operating system drive is C:. The location of the program is D:\Any_Folder\any_program.exe  How do I start/execute that program via command prompt in Windows 8?  I have tried the command line START any_program.exe, but it shows me an error that      Windows cannot find 'any_program.exe'. Make sure you typed the name correctly, and then try again.   By the way, it worked perfectly in Windows 7. And, if I type START notepad.exe or START firefox.exe (Firefox is not installed in C: drive), it works in Windows 8. Can I run a natural gas dryer from a portable propane tank? Where did Hogwarts get money for salaries, food and other purchases? We know that you can't transfigure food, so you have to grow it or buy it. Ditto money.  I'm sure that aside from some medicinal plants Herbology Professor had there were no farmlands at Hogwarts. So they must have bought food (as well as potions ingredients for Snape etc...). And they needed to pay the teacher salaries.  Where did all that money come from according to canon?  Ministry funding?  Tuition?  Patent money on Dumbledore's inventions?  Philanthropic donations by Malfoys and co?  Other? Finding the Mobius Transformation that maps open unit disk onto itself Hi I am trying to find all the Mobius transformations that map unit open disk onto itself i.e., if $|z|&lt;1$ then $|f(z)|&lt;1$ where $f(z)=\frac{az+b}{cz+d}$. I did so far \begin{align*} &amp;\Big|\frac{az+b}{cz+d}\Big|&lt;1\\ \Rightarrow &amp; |az+b|&lt;|cz+d|\\ \Rightarrow &amp; |az+b|^2&lt;|cz+d|^2\\ \Rightarrow &amp; |az+b||\bar{a}\bar{z}+\bar{b}|&lt;|cz+d||\bar{c}\bar{z}+\bar{d}|\\ \Rightarrow &amp; |a|^2|z|^2+|b|^2+2\text{Re }(az\bar{b})&lt;|c|^2|z|^2+|d|^2+2\text{Re }(cz\bar{d}) \end{align*}   After that I am stuck. Can anyone help me. I would be obliged...Thanks in advance. Where can I find Obsidian in the Pocket Edition? I have looked everywhere for lava or obsidian in 0.7.1. I've looked all the way from top to bottom and all around, in about a 100 block diameter, but I can't find any. Where can I get some? Creating compressed GeoTIFF with QGIS Clipper tool? I have a geo-tiff created by Pix4D which is about 375 Mb in file size.   I have clipped it to a smaller area using Qgis 2.8(using Raster ‣ Extraction ‣ Clipper), and the output file is about 900 MB in size.  Why did the clipping process generate such a large file, when the source file was smaller?   How do I run the clipper tool so that it produces a smaller size? I can buy Dell Latitude E7270 through my employer. It will come with Windows, but I would like to install Ubuntu (most likely 14.04). This model does not appear on the list of certified hardware, however. Does anyone know whether I should anticipate any problems with his laptop? Thanks! How force Chrome to use google.com search engine without https Chrome's default search engine is .  How we can force chrome to use  without https?  I need a solution that allows proxies to rewrite my URLs, but https currently makes this impossible... "The resource cannot be found." error when there is a "dot" at the end of the url My fiancée and I are looking for a good Caribbean cruise in October and were wondering which islands are best to see and which Cruise line to take?  It seems like a lot of the cruises don't run in this month due to Hurricane season so I'm looking for other good options.  EDIT We'll be travelling in 2012. I have a Alcatel onetouch model 50540 running android 5.1.1 as in the title. it is rooted and began uninstalling pre-installed bloatware on my device. I do not have a complete list of uninstalled apps because I had only kept track after the first few, but ill list the ones I do know about    Download Manager Downloads Talkback Hangouts Cricket Wi-Fi Manager Camera Email Onetouch Launcher   That is the essence of what I did. I do have an edited hosts file as a makeshift adblock.  Play store gives me an error code 0 to which I removed app data and tried again, same problem.  I have tried installing local apps which brings up an "app not installed" box.  Lastly I have tried do directly extract the app into /data/app and rebooting which didn't work. What translation occurs when writing to a file that was opened in text mode that does not occur in binary mode? Specifically in MS Visual C.  unsigned char buffer[256]; for (int i = 0; i &lt; 256; i++) buffer[i]=i; int size  = 1; int count = 256;   Binary mode:  FILE *fp_binary = fopen(filename, "wb"); fwrite(buffer, size, count, fp_binary);   Versus text mode:  FILE *fp_text = fopen(filename, "wt"); fwrite(buffer, size, count, fp_text); It's time for drag and drop! I often find myself wishing SO supported drag and drop. For example I drag a tag to my favs and it would add it, or when searching by tag I would like to drag the tag to the search box and get [asp.net] added in the search box.   Does it make sense to add this feature or is it just too much work for so little use? How can I produce exercises in one part of a LaTeX document with selected answers in another? I could, of course, simply typeset the questions and the answers separately, but this is too likely lead to divergence in the labels for the questions and the answers as the document is edited.   What I'd really like is a command that would allow me to typeset exercises at the point at which the command was used and which took an optional argument for a solution to be stored until a final section (chapter, or part) "Selected Exercise Answers". I'd like the questions (and the selected answers) to be labeled in the output by Chapter.Section.Question or even Chapter.Question. In the 90's I read a book about two children (I think they were siblings, boy and girl) who climbed a tree. At the end they found a door to another world. After some time the (fantasy) worlds switched to another one. Once they got stuck in a world, because the world went on traveling and therefore the door couldn't be reached anymore. They met many curious individuals during their adventures.  It's a completed novel for kids. I've read it in German, but I can't remember if the author was a German one, so it might've been available in English as well. It was a fairly thick book.  EDIT: After some thought I remember a pancake kind of person Maximal ideals of polynomial rings in infinitely many variables Let $k$ be an algebraically closed field. Nullstellensatz states that the maximal ideals of the polynomial ring $R=k[X_1,\dots,X_n]$ are precisely those of the form $\langle X_1-a_1,\dots,X_n-a_n\rangle$, with $(a_1,\dots,a_n)\in k^n$.     What if we are working with infinitely many variables, say $\{X_i\}_{i\in I}$, being $I$ an infinite set? Again, the ideals of the form $\bigl\langle\{X_i-a_i: i\in I\}\bigr\rangle$, with $(a_i)_{i\in I}\subseteq k$ are maximal (even if $k$ is not algebraically closed), but what about the converse?