What are the possible eigenvalues of a projection matrix? I know that a projection matrix is any square matrix $A$ such that $A^2 = A$. Thus, we obtain  $$ A^2v = \lambda^2 v \\ Av = \lambda v $$  for some vector $v \neq 0$. The above imply that $\lambda v = \lambda^2 v$ so $\lambda^2 - \lambda = 0$ so $\lambda = 0, 1$. Does this seem right? For some reason I feel like $-1$ should also be an eigenvalue.
A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator.   In this question is it necessary to define what the projection operator is? And won't the eigenvalue just be zero?
alias or bash function does not work When I create  alias wd='ps -ef | grep java | awk {'print $2 " " $9'} | egrep "(A|B|C|D)"'   or  function wd () {     ps -ef | grep java | awk '{print $2}' ... }   in my .bashrc file, I get errors.  Interestingly, if I source my .bashrc file with the function, it 'compiles', but when executing, gives me:   context is      &gt;&gt;&gt; \ &lt;&lt;&lt; {\print     missing }   Can someone help me with this, and also answer when its better to put something in a function versus in an alias? Essentially, I have a csv file that contain multiple columns, called cols.csv  1,a,100 2,b,200 3,c,300 4,e,400   and I have a new csv file that has one column, called col.csv  f g h i   I want to copy the items in col.csv and append them to the end of each line in cols.csv so that, cols.csv now contains those  1,a,100,f 2,b,200,g 3,c,300,h 4,e,400,i   is that possible at all? i tried join, paste, nothing worked How would I prove that if an integer $n>2$, then $\bar{z}=z^{n-1}$ has $n+1$ solutions If $n$ is an integer and $n&gt;2$, then $$\bar{z}=z^{n-1}$$ has $n+1$ solutions. Does this have something to do with the rational root theorem? Answer draft is lost if you click to sign up How to convert an "object" into a function in JavaScript? JavaScript allows functions to be treated as objects--if you first define a variable as a function, you can subsequently add properties to that function.  How do you do the reverse, and add a function to an "object"?  This works:  var foo = function() { return 1; }; foo.baz = "qqqq";   At this point, foo() calls the function, and foo.baz has the value "qqqq".  However, if you do the property assignment part first, how do you subsequently assign a function to the variable?  var bar = { baz: "qqqq" };   What can I do now to arrange for bar.baz to have the value "qqqq" and bar() to call the function? Partial SSL in rails Primefaces growl is not working in all the pages I try to create a common extent for rasters in QGIS 2.99. Here is my script:  import processing  processing.runalg("script:unifyextentandresolution",                   "/qgis_data/rasters/Image2.tif;                   /qgis_data/rasters/Image1.tif",                  -9999,"/qgis_data/rasters",True)   However, processing.runalg is being used for QGIS 2.99. I need it for QGIS 2.99 Calculating Mean for Multiple Rows in ArcGIS for Desktop? I am trying to calculate a mean across a multiple number of attribute rows and have this value added to a new field that I have entered in the attribute table. I have an attribute table which has a number of different ecoregion types associated with them.  For example:  If I select "Arid Canyonlands" there are 3 attribute rows that are selected, each with their own "mean". I am trying to add a new field to the attribute table which will take the "mean" of these three and populate that value in the new field created. I could manually do it through an editing session and using a calculator but just wanted to know if there is an easier way to actually do it through field calculator? To install the graphics driver I need to exit GUI / Gnome and go to the terminal. How can I do this in Ubuntu 18.04? Ctrl(STRG)+Alt+F1...F7 does not work, neither on the desktop, nor on the login screen... Today is the release of Ubuntu 21.04 (Hirsute Hippo). And I can't update to this version. And also I selected ‘Notify me of a new Ubuntu version’ and changed it to ‘For any new version' on the Updates tab on Software &amp; Update, but it's still not working. In addition, in the terminal, I've written: sudo do-release-upgrade -c  And output was: Checking for a new Ubuntu release No new release found.  What should I do? Proof by induction of summation inequality: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$ Can I look at the upgrade log after a distribution upgrade? I've upgraded to Ubuntu 12.04 and did it . I didn't look at the "packages to remove" section before clicking next - I've done Ubuntu upgrades many times and assumed it was just the normal set of obsolete library packages etc. But I do tend to watch the terminal text go by and see what it's doing, and I saw it removing a number of packages I wanted. I guess these are packages that aren't on the CD that can't co-exist with the new Ubuntu.  Anyway, I want to find which packages have been removed. So does the update log get saved somewhere?  Then I can start doing some grepping on it and then reinstall the packages I did want. I'm trying to create globally-unique identifiers in JavaScript.  I'm not sure what routines are available on all browsers, how &quot;random&quot; and seeded the built-in random number generator is, etc. The GUID / UUID should be at least 32 characters and should stay in the ASCII range to avoid trouble when passing them around. Relativity and Galaxy Rotation Speed If time travels slower nearer gravity wells, why can't the galaxy rotation speeds being faster on the outer edges than the inner areas be explained by relativity? What necessitates dark matter? Solve the recurrent relation How did Sauron remain in control of the Nazgûl? The One Ring gave Sauron control of the Nine, yet when it was taken from him he still possessed the power to control them (or, at least he regained it towards the end of the third age).  Why is this?   mentions that he drew the Nine to him, but not how. How to make Table of Contents manually I'm trying to make table of contents manually, I mean I set the page of each of sections one by one. Here's my target:  and I'm using book class here.  Here's the code that I found   \documentclass[oneside]{book} \renewcommand\contentsname{CONTENTS} \renewcommand\listtablename{LIST OF TABLES} \renewcommand\listfigurename{LIST OF FIGURES} \addtocontents{toc}{{\bfseries Contents\hfill Page No.\bigskip\par}} \addtocontents{lot}{{\bfseries No.\hfill Title\hfill\hfill Page No.\bigskip\par}} \addtocontents{lof}{{\bfseries No.\hfill Title\hfill\hfill Page No.\bigskip\par}}  \usepackage[titles]{tocloft} \usepackage{xcolor,xpatch} \renewcommand\cftbeforechapskip{0ex} \renewcommand\cftchapfont{\mdseries} \renewcommand\cftchappagefont{\mdseries} \renewcommand\cftchappresnum{Chapter~} \renewcommand\cftchapaftersnum{. } \newlength\tocindent \settowidth\tocindent{\cftchapfont\cftchappresnum9\cftchapaftersnum} \edef\cftchapnumwidth{\the\tocindent} \edef\cftsecindent{\the\tocindent} \advance\tocindent2.3em \edef\cftsubsecindent{\the\tocindent} \advance\tocindent3.2em \edef\cftsubsubsecindent{\the\tocindent} \renewcommand\cftsecdotsep{\cftnodots} \renewcommand\cftsubsecdotsep{\cftnodots} \renewcommand\cftsubsubsecdotsep{\cftnodots} \newcommand\tocmainmatter {\renewcommand\cftchappagefont{\color{white}}% } \xapptocmd\mainmatter{\addtocontents{toc}{\protect\tocmainmatter}}{}{}   \newif\ifschaptertoc \makeatletter \renewcommand\@makeschapterhead[1]% {{\parindent \z@ \centering     \normalfont     \interlinepenalty\@M     \Large \bfseries  #1\par\nobreak     \vskip 20\p@ }% \@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}% \ifschaptertoc \addcontentsline{toc}{chapter}{#1}% \fi } \makeatother  \usepackage{blindtext}  \begin{document} \frontmatter  \tableofcontents \schaptertoctrue  \chapter*{ACKNOWLEDGEMENTS} \blindtext  \chapter*{ABSTRACT} \blindtext  \listoftables  \listoffigures  \chapter*{ABBREVIATIONS} \Blindtext  \chapter*{NOTATIONS} \Blindtext  \mainmatter  \chapter{INTRODUCTION} \blindtext  \section{General Background} \blindtext  \section{Objective} \blindtext \subsection{Objective 1} \blindtext \subsection{Objective 2} \blindtext \subsection{Objective 3} \blindtext  \section{Scheme} \blindtext \end{document} Fantasy book with shape shifting dragon warriors Set in a fantasy world with shape shifting Dragon kings and their warriors. They terrorize the humans, and kill those with magic.   The protagonist is the great-grandson of a powerful warlock. He finds out that he has magic, and fights with it to save his new friends, the Griffin, Blackhorse (I think that's his name), some creepy necromancer who is immortal, and Gwen (fellow sorceress), along with the kingdom that the Griffin rules in.   They fight the protagonist's father, the leader of the Dragon kings, the king's Duke, and the creepy sorcerer, who has a curse (when he dies, he changes side from good to bad, or bad to good). I have no clue what this is called, but I want to find out, since I'm out of things to read. Which Defiant was shown in ST: First Contact? In Memory Alpha, the page on the  states that two Defiant-class ships had that name: the prototype Defiant, and a successor, the U.S.S. Sao Paulo, which was assigned to DS9 and renamed Defiant after the loss of the original. All of this happened during the DS9 series.  In the Star Trek: First Contact movie, the Defiant participates in the opening battle, where a Borg Cube penetrates the Sol system's defenses and besieges Earth. Worf is in command, and is transported out of the crippled Defiant along with whatever's left of his crew by the Enterprise; out-of-universe, this is a rather transparent method to get Worf back into the TNG side of things after the events of ST: Generations, when he moved to DS9 as tactical officer and unofficial envoy to the Klingon Empire.  The question is, was Worf's Defiant in the movie the original, or the Sao Paulo? This would primarily depend on where the movie falls in the overall canon timeline. According to in-movie dialog, whichever one it was was salvageable and therefore would have survived the movie, so the movie would have had little impact on the canon of any running TV series at the time.