After calculating the eigenvalues, I get $1040400$ and $0$. Since one of them is $0$, how do I calculate (orthogonal) matrix $U$? $$u_i = \frac{1}{\sqrt{\lambda_i}} A v_i$$ I know there are a similar question here, but when calculation $u_1$ it's a null vector.  So, I don't see how I can find $u_2$.
The question is if you have a situation where one of the singular values is equal to 0 in a singular value decomposition of a matrix, how to do you procede to find the column vector of U corresponding to this singular value?   Usually I found the columns of U with this relation, so let's say that we refer to $\vec{u_2}$ as the U column vector corresponding to singular value equal to 0:  $\vec{u_2} = (1/s_2)Av_2$  Now the as we can see the 0 is in an illegal place, since you can't divide by zero. But if you move it to the left side, I still can't quite see how to find the column of U corresponding to it. Any suggestions?  Here is an example of a matrix that provided this specific challenge  $ A = \left(\begin{array}{rrr} -3 &amp; 1 \\ 6 &amp; -2 \\ 6 &amp; -2 \end{array}\right). $  $ \lambda{_1} = 90$,  $\lambda{_2} = 0$,  $s_1 = 3 \sqrt{10}$,  $s_2 = 0$,  How can one find $\vec{v_2}$ when $\lambda{_2} = 0$? In this case, you know that the eigenvectors of $A^{t} A$ are orthogonal, solve for $\lambda{_2}$ and find the eigenvector if it's orthogonal to the eigenvector corresponding to the first eigenvalue you have your solution.  So we end up with the following normalized column vectors for V, which we can see are orthogonal:  $ \vec{v_1} = \left(\begin{array}{rrr} -3/\sqrt{10} \\ 1/\sqrt{10} \\ \end{array}\right). $  $ \vec{v_2} = \left(\begin{array}{rrr} 1/\sqrt{10} \\ 3/\sqrt{10} \\ \end{array}\right). $
How to smooth shade an object while retaining hard edges? Is there any adjustment required when computing the correlation coefficient between:   continuous and discrete variable  two discrete variables ?    Thanks ! I am running Ubuntu 15.10 with Gnome DE.  I installed KDE plasma to try it out but sudo apt-get upgrade started giving me errors so I ppa-purged the kde ppa and removed kubuntu desktop with the terminal in Gnome.  Now when I run sudo apt-get upgrade , I get the following error:    The following packages have unmet dependencies:     kde-telepathy-minimal : Depends: kde-config-telepathy-accounts      (&gt;= 15.04.0) but it is not installed.   sudo apt-get install -f does not resolve the issue. I tried sudo apt-get remove kde-telepathy-minimal to remove the package which gives me this error:    The following packages have unmet dependencies: kde-telepathy : Depends: kde-telepathy-minimal   Attempting to remove 'kde-telepathy' gives me the original error.  I am stuck. Please help me and thank you Ubuntu community! Android : Is there any alternative to listFile function (Android java.io.File.listFiles()) Random spawn location despite sleeping in a bed I am trying to make Minecraft harder, and one of the ways I wish to do this is to have a "pseudo-hardcore" mode where dying spawns you in a random location rather than completely locking you out of the world.  I intend to do this by setting the spawn radius to be very high.  However, I also want to make this "bed-proof" - that is, if someone places a bed and sleeps in it, I want the player to still be spawned in a random location, regardless of their personal spawn point.  How do I do this? Command blocks are fine, but data packs are not preferred. According to the documentation, the  method uses a round-to-even algorithm which is not common for most applications. So I always end up writing a custom function to do the more natural round-half-up algorithm:  public static decimal RoundHalfUp(this decimal d, int decimals) {     if (decimals &lt; 0)     {         throw new ArgumentException("The decimals must be non-negative",              "decimals");     }      decimal multiplier = (decimal)Math.Pow(10, decimals);     decimal number = d * multiplier;      if (decimal.Truncate(number) &lt; number)     {         number += 0.5m;     }     return decimal.Round(number) / multiplier; }   Does anybody know the reason behind this framework design decision?  Is there any built-in implementation of the round-half-up algorithm into the framework? Or maybe some unmanaged Windows API?  It could be misleading for beginners that simply write decimal.Round(2.5m, 0) expecting 3 as a result but getting 2 instead. Each site got a &quot;hot&quot; tab, where its contents are calculated based on . However take a look on the screenshots below and see for yourself that &quot;hot&quot; got nothing to do with many of the questions listed there. :    :    As you can see, there are questions with negative score, no answers, and even closed. I won't be surprised to see spam in there as well. This can't be any worse. What I suggest, instead of fixing the formula or patching it, simply have the hot tab show the same questions as the Hot Network Questions sidebar. (from the specific site only, of course.) Since the hot questions &quot;pool&quot; (as seen ) is limited to 100 for the whole network, either expand it or have the per-site list show questions with less hotness points - it would still be better than what we have now. Books on cosmology I have finished Carroll's general relativity book, Spacetime and Geometry. I am specifically interested in cosmology, so is there any book which goes more in depth into cosmology? I prefer a mathematically rigorous book, which goes beyond the cosmology chapter in Carroll's book. The Big Bang and inflation theory should be dealt with in depth. Possible Duplicate:          I am always confused with the prepositions to use when indicating an event happening at a place. Should I use "at" or "in" or "on"?  For example:     We had our dinner at the McDonalds at/in the shopping centre at/on Sixth   Avenue.   The McDonalds restaurant is in the shopping centre. The shopping centre is located at Sixth Avenue. So am I right to say "the McDonalds in the shopping centre at Sixth Avenue"?  Now, what if it was a street instead of an avenue? Say for example:     We had our dinner at the McDonalds at/in the shopping centre at/on Coral Street.   Here, somehow, I find it natural to say "the McDonalds at the shopping centre on Coral Street".  How do I determine the right prepositions to use? And how would the different prepositions give different meanings to the sentence? Can you concentrate on a special ability while also concentrating on a spell? There exist several abilities that have the text similar to, this effect lasts while you maintain concentration (as if you were concentrating on a spell). One such example is the Bard College of Glamour: Mantle of Majesty ability.     At 6th level, you gain the ability to cloak yourself in a fey magic   that makes others want to serve you. As a bonus action, you cast   command, without expending a spell slot, and you take on an appearance   of unearthly beauty for 1 minute or until your concentration ends (as   if you were concentrating on a spell). During this time, you can cast   command as a bonus action on each of your turns, without expending a   spell slot.   Another example is the Cleric Trickery Domain: Channel Divinity: Invoke Duplicity ability.     Starting at 2nd level, you can use your Channel Divinity to create an   illusory duplicate of yourself.      As an action, you create a perfect illusion of yourself that lasts for   1 minute, or until you lose your concentration (as if you were   concentrating on a spell).   While you are concentrating on maintaining such an ability would you also be able to maintain concentration on a spell that requires concentration?  There does not seem to be any RAW that limits this however the limitation around concentrating on a second spell exists which may suggest that the intent is that a character cannot concentrate on 2 effects at the same time unless explicitly stated. Text formatting now allowed in comments (List of proven and disproven abilities) How can I prove that the Cartesian product of two countable sets is also countable? I know it is defined as $ dP \wedge dx  $ or explicitly,  $$ \frac{\partial P }{\partial y } dy \wedge dx   . $$  The question is, could it be $dx \wedge d P $?  Or  $$  \frac{\partial P }{\partial y }  dx \wedge dy ?    $$  I know the question must be very naive or even stupid. But I am indeed confused. Any nice, stable ways to keep a window 'Always on top' on the Mac? I am looking for a stable way to keep any window 'always on top' on OSX.  , an app that does that requires SIMBL/easySIMBL, which it seems is not very stable.  Are there any better solutions? It's actually surprising that there is no native way to implement this. While watching  play for the gold medal in London 2012, I noticed Williams was going about her business while Sharapova was grunting each time she hit the ball.  After doing some , grunting in tennis dates back to 1962. Although men grunt, women draw more attention due to their high-pitched grunts. In addition, there is an initiative to eliminate excess grunts in the next generation of tennis players.  Another quote from said research says:     "I have not found in my 32 years anybody who says grunting or not   grunting has an empirically positive or negative effect on performance   -- it's all about belief system."   What is this "belief system?"  If grunting is not related to performance, why do tennis players do it excessively? How do you check for whether a device is in a wifi deadzone? I am completely new in Ubuntu. Just installed Ubuntu 14.04. On my other computer running Windows, I am using Avast, free edition.  Do I need anti-virus protection at all? Is it possible to double your proficiency bonus twice? Let's say I'm a multiclass rogue/ranger and I have taken Expertise in survival checks. Now, if a ranger is in their favored terrain it says they double their proficiency bonus in intelligence and wisdom checks relating to you environment. If I am in my favored terrain and use my already doubled survival check, will it double again? Incorrect plot using pgfplots (trigonometric functions like cos, sin and tan) Gamma Distribution out of sum of exponential random variables