Am I right that "spectral decomposition" for symmetric matrix and "singular value decomposition" for non square matrix? Any clarification would be appreciated.
I have understood how ridge regression shrinks coefficients towards zero geometrically. Moreover I know how to prove that in the special "Orthonormal Case," but I am confused how that works in the general case via "Spectral decomposition."
Diagonal lines in table cell I need this table:    (each row and column should have the same height and length, last cell should be divided by diagonal line). I tried the following code:  \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{array} \usepackage{makecell} \newcolumntype{x}[1]{&gt;{\centering\let\newline\\\arraybackslash\hspace{0pt}}p{#1}} \begin{document} \setlength{\extrarowheight}{0.1cm} \begin{tabular}{|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|}\hline &amp;&amp;&amp;&amp;20\\ \hline &amp;&amp;&amp;&amp;30\\ \hline &amp;&amp;&amp;&amp;45\\ \hline 15&amp;12&amp;18&amp;50&amp;\diaghead(-3,2){\hskip \hsize}{$a_i$}{$b_j$}\\ \hline \end{tabular} \end{document}   but text in the last cell is displayed incorrectly and cells are not the same.     How can I change that? Some three-digit (incl. minus sign) vote count still doesn't display properly Related:      As the title says...    The Android app version is 0.1.58.  Here is my experimental results:   Vote counts >= 200 that I've tried seems to be fine Vote count == 241 has problem, 242 doesn't Vote counts which matches /^1[012345689][0-9]$/ seems to be having problem (may be overgeneralized) Vote count == -36 definitely has problem   Note that this is not a tablet, but a 4-inch mdpi phone, of 480x854px. Beginner here with an issue on a simple presentation using beamer  \documentclass{beamer}  \begin{document} \begin{frame}   \tau \epsilon \chi \end{frame} \end{document}   The above fails with the message "! Missing $ inserted." and waits for me to do something when converting to pdf. I have tried adding the instruction \usepackage[utf8]{inputenc} and also \usepackage[greek]{babel}. I have not seen references either to needing to use any other package. What is exactly I am missing? Does applying the Collatz conjecture algorithm to negative integers always result in sequences ending with $-1$ and then repeating themselves an infinite number of times? It is so for the first several seed values.  After hours of thinking and research, I don't know how to answer the question. Merge two custom post types into one admin page? I am building a site with 4 custom post types. To make things easier for our client/admin, I'm wondering if there is a way to merge those custom post types into one list in the backend? Like if you look at normal backend admin page with the list and columns of posts, can a page like that pull in 3 other post types for easy sorting/searching?  I can't combine the CPTs into one, they must remain seperate outside of one 'viewing' page in the admin area if possible.  Any thoughts on how to do this?? Or even where to start looking/thinkging of how to develop this? Two diagrams side by side I am trying to put two feynman diagrams side by side, but they get displayed below eachother. I am using feynmf and subfloat. This normally works for pictures.  \begin{figure}[ht!]       \centering     \subfloat{          \begin{fmfgraph*}(40, 25)         \fmfstraight          \fmfbottom{wm,i2,z2}           \fmftop{wp,i1,z1}           \fmf{dashes}{wp,i1,z1}           \fmf{dashes}{z2,i2,wm}           \fmf{dashes,caption=$W^{+}$}{i1,i2}         %  \fmf{dashes,caption=$Z^0$}{i2,z1}          % \fmf{dashes,caption=$Z^0$}{i2,z2}         \fmfv{lab=$Z^0$}{z1}         \fmfv{lab=$Z^0$}{z2}     \fmfv{lab=$W^+$}{wp}     \fmfv{lab=$W^-$}{wm}         \end{fmfgraph*}     }      \subfloat{          \begin{fmfgraph*}(40, 25)         \fmfstraight          \fmfbottom{wm,i2,z2}           \fmftop{wp,i1,z1}           \fmf{dashes}{wp,i1,z1}           \fmf{dashes}{z2,i2,wm}           \fmf{dashes,caption=$W^{+}$}{i1,i2}         %  \fmf{dashes,caption=$Z^0$}{i2,z1}          % \fmf{dashes,caption=$Z^0$}{i2,z2}         \fmfv{lab=$Z^0$}{z1}         \fmfv{lab=$Z^0$}{z2}     \fmfv{lab=$W^+$}{wp}     \fmfv{lab=$W^-$}{wm}         \end{fmfgraph*}     }      \caption{Exchange of W boson}     \end{figure} How to compute rational or integer points on elliptic curves This is an attempt to get someone to write a canonical answer, as discussed in this . We often have people come to us asking for solutions to a diophantine equation which, after some clever manipulation, can be turned into finding rational or integer points on an elliptic curve. See , , , , . (This list is biased towards questions I have answered because I remember them best; other people have answered such questions as well.) I would like an answer to which, after one of us has explained the cleverness, we could direct the OP.  An ideal answer would address    How to find both rational and integer solutions Good software solutions. Ideally, it would nice to have a walkthrough for doing these things with , so that people could find solutions without even installing anything. References for how to transform some standard presentations of elliptic curves into Weierstrass form, so that we don't have to write out the algebra every time. I'm thinking of  a cubic in $\mathbb{P}^2$ with a rational point that is not a flex,  $y^2 = \mbox{degree 4 polynomial}$, a $(1,1)$ curve on $\mathbb{P}^1 \times \mathbb{P}^1$, or an intersection of two quadrics in $\mathbb{P}^3$. I bought a new MacBook Air and was able to connect to the internet at first but now it won't connect.  I have other devices as well as other computers on wifi but my Mac won't connect.  When I try and click on the wifi, it tells me that it can't be joined.  How can I troubleshoot? Probability that the sum of 6 dice rolls is even L293, L298 and SN754410 H-bridge drivers on low voltage power supply I want to log out of Stack Overflow or any Stack Exchange site, but always it shows me logged in!  How do I log out of this site?  Note: I have yet to register; the site currently always shows my name and default email. I wish to log out so that I may start a new session! jQuery doesn't work after content is loaded via AJAX Why does polynomial factorization generalize to matrices I'm reading about linear algebra and I came across with the following theorem where I have a problem convincing myself:      Theorem 2.1 $\,$ Every linear operator on a finite-dimensional complex vector space has an eigenvalue.      Proof:       To show that $T$ (our linear operator on $V$) has an   eigenvalue, fix any nonzero vector $v \in V$. The vectors $v, Tv,  T^2v,..., T^nv$ cannot be linearly independent, because $V$ has   dimension $n$ and we have $n + 1$ vectors. Thus there exist complex   numbers $a_0,...,a_n$, not all $0$, such that      $$a_0v + a_1Tv + ··· + a_nT^nv = 0.$$ Make the $a$’s the coefficients   of a polynomial, which can be written in factored form as $$a_0 + a_1z  + ··· + a_nz^n = c(z − r_1)\cdots(z − r_m),$$ where $c$ is a non-zero complex number, each $r_j$ is complex, and the equation holds for all   complex $z$. We then have      $${\color{red}{ 0=(a_0I + a_1T + ··· + a_nT^n)v= c(T − r_1I)\cdots(T −  r_mI)v}},$$ which means that $T − r_j$ I is not injective for at least   one $j$. In other words, $T$ has an eigenvalue.$\;\blacksquare$   I'm having trouble with the factorized form of the matrix polynomial (in red). I understood that the factorization holds for a polynomial by the fundamental theorem of algebra but why does this also hold for matrices?  In other words, why is the the part I highlighted true? Does such an factorization always exist?    Could I have some help to see this? Thank you =)   P.S.  is my reference (page 3).  UPDATE:   Someone else also has asked the  before it seems. "Unable to locate package" while trying to install packages with APT When I try to install any package through the command line, I get an error.   $ sudo apt-get install &lt;package&gt; Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package &lt;package&gt;   Can anyone help me on this? Is Using Digital Signatures to prove identity a zero knowledge proof? Suppose Alice publishes a book with a public key in it, and later wants to prove that she wrote the book. She could sign challenge messages with her private key, and others could verify those signed messages. Alice is proving she knows a private key without revealing anything about it.  Is this an example of an interactive zero knowledge proof?   Or does this example somehow not apply to zero knowledge proofs? Why can't $ i\hbar\frac{\partial}{\partial t}$ be considered the Hamiltonian operator? I was wondering how to run a command as another user from a script.  I have the script's owner set as root. I also have the following command being run within the script to run the command as the hudson user:  su -c command hudson   Is this the correct syntax? I'm looking for the name of a series of space opera novels I read some time ago, probably between 2003 and 2010.    It may have been published quite a few years before I read it, I don't think it was new at the time. I borrowed the novels from the local library, but I can't find them in its catalogue. I think the covers were fairly dark and plain, though there may have been some artwork on them. There was a totalitarian empire that controlled most planets, possibly ruled by a queen. There was no mention of Earth, the solar system or other real planets, the setting was completely fictional. There were people with psychic/psionic powers, referred to as something like "psi". These people were second-class citizens or slaves. Spaceships were used to travel between worlds. I'm not sure what kind of FTL technology they had, if any. Beyond the edge of the empire was some kind of "dark space", where there was no light. There were some horror themes: When travelling in dark space in one of the novels, a ships crew experienced strange thoughts and hallucinations. I remember finding this quite scary at the time.   Does anyone know which series this could be?  This isn't a duplicate of , since the two questions are about different parts of the same series (the first and second episodes). Can't install libglew-dev because libcheese and libclutter don't have the required versions I'm getting the following error message when attempting to install libglew-dev,  Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  libcheese-gtk23 : Depends: libclutter-gtk-1.0-0 (&gt;= 0.91.8) but it is not going to be installed                    Depends: libcogl15 (&gt;= 1.15.8) but it is not going to be installed  libcheese7 : Depends: libclutter-gst-2.0-0 (&gt;= 0.10.0) but it is not going to be installed               Depends: gstreamer1.0-clutter but it is not going to be installed  libclutter-1.0-0 : Depends: libcogl-pango15 (&gt;= 1.15.8) but it is not going to be installed                     Depends: libcogl15 (&gt;= 1.15.8) but it is not going to be installed E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.   All those packages are however apparently already installed.  I have tried all possible solutions in  post, without any luck. I have not added any PPAs. The version of ubuntu is 14.04.2.  I have looked at similar posts, and tried the following:  Running  dpkg --get-selections | grep hold   prints nothing.  sudo apt-get -oDebug::pkgProblemResolver=true install -f   Gives me nothing of interest, 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. When I'm looking down the front page on , I often wish I could tell at a glance what happened to the question to bump it there. Sometimes it is obvious from looking at the number of answers, other times you have to follow a link to find out if it has been edited or if an answer has been added or updated.  I'm guessing this is just as applicable on other SE sites - this is the kind of thing I'm suggesting:    A couple of suggestions from the comments that might be good:   To reduce clutter the word 'ago' could be removed at the same time. The new text could be visible only to established users - ie users who are likely to be reviewing/editing and would benefit from seeing at a glance what kind of change has occurred.