Given $\mathbf v=\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}$ and $\mathbf A=\mathbf{vv}^\top$, find the eigenvectors and eigenvalues of $\mathbf A+\lambda\mathbf I$.   My current work progress is:  Since $(\mathbf A+\lambda\mathbf I)\mathbf v =\mathbf{Av}+\lambda\mathbf v=\mathbf v(\mathbf v^\top\mathbf v) + \lambda\mathbf v=\mathbf v \|\mathbf v\|+ \lambda\mathbf v=(\|\mathbf v\|+\lambda)\mathbf v$ ,  so one of the eigenvalue is $\lambda_1=(\|\mathbf v\|+\lambda)$ with corresponding eigenvector $\mathbf v_1=\mathbf v$.  I'd like to find out other pairs of eigenvalues and eigenvectors. I first calculate the trace.  $$\mathrm{Tr}\,(\mathbf A)=n\lambda+\|\mathbf v\|=\lambda_1+(n-1)\lambda$$  Hence, the rest of $n-1$ eigenvalues has sum $(n-1)\lambda$.   From here, I do not know how to proceed.   Can I assume the rest of eigenvalues are $\lambda$ with multiplicities $n-1$? Why? How to find corresponding eigenvectors?   [Note1] This is like a follow up question regarding .  [Note2] Found later on there is a  but people focus on linking it to Note1.
This question aims to create an &quot;&quot;  of numerous questions that ask about determinants of specific matrices (I may have missed a few):              The general question of this type is  Let $A$ be a square matrix of rank$~1$, let $I$ the identity matrix of the same size, and $\lambda$ a scalar. What is the determinant of $A+\lambda I$?  A clearly very closely related question is  What is the characteristic polynomial of a matrix $A$ of rank$~1$?
I have recently upgraded Ubuntu 19.10 to latest LTS Ubuntu 20.04. After updating, I get serious artifacts on my screen. Every time I reboot (or log-in), my screen is fuzzy or scrambled (picture attached):     This is temporary. The fuzzy screen goes away when I change the background. But it reappears the moment I reboot or log back in.   I have Intel HD Graphics 520 (Skylake GT2).  The output of lspci -nn |grep -E 'VGA|Display' is:  00:02.0 VGA compatible controller [0300]: Intel Corporation Skylake GT2 [HD Graphics 520] [8086:1916] (rev 07)   Update:   On enabling Wayland in Ubuntu 20.04 and logging in to, these artifacts are however cured.   This is okay for now, but I am not sure about Wayland as I have not yet used it before.   Is there a way to correct the graphic issue with normal Ubuntu startup? Distance metric for (infinite) lines in 3D I would like a metric $d(\;)$ between pairs of (infinite) lines in $\mathbb{R}^3$, with these properties:   If two lines $L_1$ and $L_2$ are parallel and separated by distance $x$, then $d(L_1,L_2) = x$. $d(L_1,L_2)$ increases with the degree of skewness between the lines, i.e., the angle $\theta$ between their projections onto a plane orthogonal to the shortest segment connecting them (dashed below), where $\theta=0$ for parallel lines and $\theta=\pi/2$ for orthogonal lines. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;   Intuitively, I would like the metric to be related to the repulsive force between two electrically charged lines (but it need not be exactly the actual physical force).  Have such metrics been considered in the literature? If so, I'd appreciate descriptions and/or pointers&mdash;Thanks!  Update1. It turns out that the natural ad hoc definition (from the comments) fails to be a metric. For example, below, a sufficiently large value of $a$ ensures the triangle inequality will be violated: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; So this question may be more difficult than it initially appeared...  Update2.  The  have revealed that in fact there is no metric that satisfies my two conditions (and is continuous w.r.t. $\theta$)! Is there an invalid pthread_t id? Why is gradient the direction of steepest ascent? $$f(x_1,x_2,...x_n):\mathbb{R}^n \rightarrow \mathbb{R}$$ The definition of the gradient is  $$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\  ... +\frac{\partial f}{\partial x_n}\hat{e}_n$$  which is a vector.  Reading this definition makes me consider that each component of the gradient corresponds to the rate of change with respect to my objective function if I go along with the direction $\hat{e}_i$.  But I can't see why this vector (defined by the definition of the gradient) has anything to do with the steepest descent.  Why do I get maximal value again if I move along with the direction of gradient? The link to the recommended job in the app brings you to a page with the text: Not Found    Looking at the link it seems like after changing the URL structure in Stack Overflow Jobs they have not been updated in the apps.  For the job in the screenshot this is the used URL in the app:    The correct URL is: Where can I buy an Ubuntu Phone? I need to buy an Ubuntu phone, where can I buy it? It is unavailable everywhere. What temperature is achieved in focus point by 5000 flat 1x1cm mirrors onto a satellite dish? There is this  ("R5800 Solar Death Ray") where teenager built a 5000 mirror device which concetrates the solar rays which is showing the potency of the mosaic method of concentrating sunlight using elliptical dish. It can fire the wood, boil the water in seconds, melt an aluminum, glass or even burn the whole in concrere or rock.  What is the approximate maximum temperature (could be a range) is achieved by this mirror device in its focus point on sunny day?       The R5800 solar concentrator, replaced by the new R23k, is made from   an ordinary fiberglass satellite dish. It is covered in about 5800   3/8" (~1cm) mirror tiles. When properly aligned, it can generate a   spot the size of a dime with an intensity of 5000 times normal   daylight. This intensity of light is more than enough to melt steel,   vaporize aluminum, boil concrete, turn dirt into lava, and obliterate   any organic material in an instant. It stands at 5'9" and is 42"   across.   Technical details &amp; calculations:     Calculations: Area of dish aperture (elliptical)= (pi x 102cm x   73cm)/4 = 5848 sq cm       Area of focal point (circular)= pi(0.6cm)^2 = 1.14 sq cm      Dish Type= 0.7m x 1m offset focus elliptical.      Concentration Power= 5000x      Output Power Estimate = 560 watts Why didn't Harry die in the dark forest? Testing a UPS by unplugging it from the wall At an Not For Profit/Charity I volunteer for occasionally we were very fortunate to be provided (donated) with a brand new 3Kva APC UPS. This is total overkill for our modest server rack (four mid-range servers and a switch!) but hey, I'll take what I can get!  Pressing the TEST button on the front indicates that yes, the UPS does work. Brilliant. But it only tests it for about 15 seconds.  My question is - will I degrade the UPS by unplugging it from the wall to see how long it will last? My plan is to unplug it, and wait until the battery meter reaches its last LED before plugging it back in, so that I know about how long I will have in the event of a power outage.  Do people do this on a regular basis? I'm guessing no (Lead Acid is very different to Li-Ion batteries)... but what kind of harm would it do if this were to happen (on purpose) every 6 months? Parallel evaluation produces wrong result Show a matrix in detail Below it is a very useful command to create a matrix. But it has a short point to me since it shows Parentheses not Square brackets. My question it how should I change this to show square brackets? I appreciate your answer. Thank you so much.  \[ K =   \underbracket{\begin{pmatrix}    K_1 &amp; 0      &amp; 0    \\    0   &amp; \ddots &amp;    0   \\    0   &amp; 0      &amp; K_n    \\    \end{pmatrix}}_{d\times M} \] Mavericks unusually high RAM usage I thought Mavericks was suppose to utilize RAM memory better, thus run smoother.  Running ML on Mac mini 2012, with 16GB RAM...always running smoothly even with over a dozen apps running.  Now, after installing Mavericks, my Mac is SLOW, eating up RAM like crazy! Looking over usage, of the 16GB of installed memory...on average it uses 10-14 GB RAM when barely running a few apps!!!   There is this kernel_task that is always running with over 1GB of RAM and most apps are not taking that much RAM, so what is using up all my RAM?! File cache? It shows very high in that new description, with about 1/3 of my total RAM What does "can only heal so much" mean in this sentence? Is there any function that converts an escaped Url string to its unescaped form? System.Web.HttpUtility.UrlDecode() can do that job but I don't want to add a reference to System.Web.dll. Since my app is not a web application, I don't want to add a dependency for only using a function in an assembly.  UPDATE: Check  about the same issue. Where can I find Vagrant packages? is a tool to build and manage virtual machines for developers.  It has a  guide that contains iffy-looking suggestions like  $ sudo ln -s /usr/bin/ruby1.8 /usr/bin/ruby  # wtf???   or installing RubyGems from source rather than from Ubuntu packages, and then using gem install to install vagrant itself.  I'm not feeling comfortable just following those instructions.  Is there perhaps a PPA?  Are there, perhaps, alternative tools that are packaged for Ubuntu? How to compile self-referencing functions (to perform recursive a task) I am trying cmd --> MMC --> Server Manager and then connecting to my Server 2012 no GUI  but i keep getting the following error The WinRM client cannot process the request  I am unable to upload images right now  This is not related to other post "How to Remote Manage Windows Server 2012 in Server Manager?" because title and description of that post is quite different from what i am asking Incorrect bookmarks and page number in table of contents I have problem with table of contents and with bookmarks. In pdf file   table of contents shows wrong page number of Introduction and Bibliography, Bookmarks: introduction and bibliography don't bring me to the correct page.   Part of my code  \documentclass[12pt,a4paper]{book}  \usepackage[polish]{babel} \usepackage{polski} \usepackage[cp1250]{inputenc} \usepackage[pdftex,pagebackref=false,draft=false,pdfpagelabels=false,pdfstartview=FitH,pdfstartpage=1,bookmarks=true,pdfauthor={LiN},pdftitle={PD},pdfsubject={Titlei},pdfkeywords={words},unicode=true]{hyperref}  \usepackage{tocloft}  \begin{document} \tableofcontents \addcontentsline{toc}{chapter}{Introduction} \chapter*{Introduction} \chapter{Chapter1} \section{Section} \addcontentsline{toc}{chapter}{Bibliography} \begin{thebibliography}{9999999}% \bibitem{1}Author, \emph{Title of book}, Publisher. \end{thebibliography} \end{document}   How can I change that? Ubuntu 14.04, problem when install gstreamer1.0-libav, unmet dependencies In Drush 8, my site aliases are loaded correctly from ~/.drush/mysite.aliases.drushrc.php.  E.g.  $aliases['dev'] = array(   'root' =&gt; '/var/www/html/mysite.dev/docroot',   'uri' =&gt; 'mysite.prod.acquia-sites.com',   'remote-host' =&gt; 'mysite.ssh.acquia-sites.com',   'remote-user' =&gt; 'mysite.dev',   'path-aliases' =&gt; array(     '%drush-script' =&gt; 'drush' . $drush_major_version,   ), );   However these aliases aren't loaded in Drush 9.  $ drush site-alias {  }   How I should define site aliases in Drush 9? Is there any migration process?