Let $T:V\rightarrow V$ be a linear map. If $\lambda^2$ is an eigenvalue of $T^2$, then $\lambda$ or $-\lambda$ is an eigenvalue of T.  I have then $(T\circ T )(v) = T^2 (v) = \lambda^2 v$, but $\lambda^2 = (-\lambda)(-\lambda)$ or $(\lambda)(\lambda)$.  Any hints
$A$ is an $n\times n$ matrix of complex numbers. Prove that if $\lambda$ is an eigenvalue of $A^2,$ then $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A.$    If $\lambda$ is an eigenvalue of $A^2,$ we have $\lambda\alpha=A^2\alpha$ for some $\alpha.$ Then how can we find a $\beta$ s.t. $\sqrt{\lambda}\beta=A\beta?$
Create a Collection inside another Collection with python How to create a Collection inside another existing Collection with Python? We roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by 6? So the question is really hard I think. I tried using a simple way by calculating the probability of each combination that makes a sum divisible by six, but it would take forever. Does anyone have any ideas?  Suppose that we roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by six? What to do with meta questions referring to currently non-existing features? I sometimes come across such questions like that:    The "Not a real question" close reason doesn't exist for long time. It doesn't make sense to answer, comment, or even up/downvote (since there's nothing more to vote for/against it).   Shouldn't such questions be locked as historical ones? Should I flag them for moderator attention or just ignore them? How can I stop Lollipop ignoring my wifi network? Android 5.0 has a new "feature": If a wifi network has no connection to the internet, it will remain connected to it but route traffic over mobile data instead. A small exclamation mark shows in the status bar over the wifi symbol.  Unfortunately, for some reason Android has decided that my home wifi does not have a good internet connection - which is not true, as I'm using it to access Stackexchange at the moment - and is, instead, trying to send everything over GPRS.  With mobile data disabled, things work over wifi just fine, despite the exclamation mark - so it isn't a problem with the wifi not working, per se.  Is there any way to override this "smart" feature? The only way I have found to make things work at the moment is to disable mobile data while I am in my house - which is not a practical long-term solution.   EDITING to add that I have now solved this problem by buying a new router! I was using something very old... But, I'll leave the question here for the benefit of anybody else with the same difficulty. Is there a word for a person who gives out too many extraneous details? $100$-th derivative of the function $f(x)=e^{x}\cos(x)$ I've got this task I'm not able to solve. So i need to find the 100-th derivative of $$f(x)=e^{x}\cos(x)$$ where $x=\pi$.  I've tried using Leibniz's formula but it got me nowhere, induction doesn't seem to help either, so if you could just give me a hint, I'd be very grateful.  Many thanks! Find total number of relations that are equivalence as well as partial order set. Assume set contains total $n$ elements.     My attempt:  As equivalence relation has property reflexive, symmetric and transitive and POSET has property reflexive, antisymmetric and transitive.  Symmetric and antisymmetric will cancle each other except reflexive relation and transitive relation can case antisymmetric, so it's also absent on resultant relation.  Therefore, all diagonal element should be present to satisfy conditions of equivalence as well as POSET.  So, it's independent of number of total number of element of set and answer will be $1$.     Can you explain in formal way, please? What happens if Superman is near a white dwarf star? With the Superman lore, we know that light from a yellow star gives him his powers, light from blue stars makes him stronger, and light from red stars weakens him.  However, what happens if he is near a white dwarf, or a neutron star?  Has there been any evidence of Superman visiting a star system that wasn't red/yellow/blue? How do you get JavaScript/jQuery Intellisense Working in Visual Studio 2008? I thought jQuery Intellisense was supposed to be improved with SP1. I even downloaded an annotated version of jQuery 1.2.6, but intellisense will not work in a separate jscript file. I have the jQuery library referenced first on my web page in the &lt;head> tag. Am I doing anything wrong? How can I install Minecraft? I have tried many tutorials but none of them are clear enough. I have also tried installing it using Wine but it won't update and Wine goes gray and stops working.  Does anybody know an easy way to install Minecraft though the Java file without using Wine? How can I make up for weak grades while applying for a masters? I feel I've received poor grades in my bachelor's degree in Mechanical Engineering - Second Class Second Division as per UK grading system. Having finished my bachelors in the UK I wish to enroll into a masters program in Systems or Mechanical Engineering in the US. I am currently employed by a logistics company in Dubai.   How can I improve my chances of getting into a grad school in the US with what I consider as poor grades? What other factors can strengthen my application? show that a distance function is continuous Let $X$ be a metric space with metric $d$. Define $d: X \times X \to \mathbb{R}$, show that $d$ is continuous.  I would like to show that the function is continuous the topology way (since it is a course on topology).   Let $(a,b)$ be a basic open set in $\mathbb{R}$, then $d^{-1}(a,b) = \{(x,y): a&lt;d(x,y)&lt;b\}$. Define $A= \bigcup_{x \in X} B_{b} (x)$, where $B_b (x) = \{y \in X| d(x,y)&lt;b$}. Clearly $A$ is open. Now define $C = \{(x,y):d(x,y)&gt;a \}.$ Essentially I hope that $A \cap C$ is open. But I am stuck in showing that $C$ is open.  Any help please? :) How to install h.264 decoder? I tried playing videos in movie player but only the audio could be heard. I tried playing the video with dragon player but here also only the audio could be heard.  A message box appeared showing the following message:     Python (v2.7) requires to install plugins to play media files of the following type: H.264 decoder   I tried installing the plugin but it couldn't be installed. It isn't available in the software center too. How can I get this plugin? Why is the [I'm Done] button unavailable after only flagging comments in the First Posts review queue? I just flagged two comments during a review of a question in the First Posts queue (on another SE site). Then I wanted to select the button I'm Done, but it's unavailable (I can only select No action needed). Even if I refresh my page (while still on that review page), it remains the same.  If however I also downvote the question (only temporary, as a test, since I don't want to downvote the question being reviewed), then it seems to work as I'd expect it to: the button I'm Done becomes available, and No action needed becomes unavailable.  Why is the I'm Done button unavailable after only flagging comments in the First Posts review queue?  Notes:   I'm not convinced that you're supposed to review the actual post, and not (only) the comments underneath it (as mentioned in 's comment). Because "if" I would post a new comment, or just upvote an existing comment, I can actually select I'm Done. While in this case I flagged existing comments which IMO had become obsolete. Flagging comments is "acting on comments". And if my flag gets accepted (in this case: "no longer needed"), and the comment is actually deleted (by the mod), then it makes the question a bit more readable (= don't waste your time digesting a comment which is obsolete). Therefor "IMO", in the end I also "acted on the post". Think about all those posts with a zillion of comments, of which many are obsolete (e.g because the post got updated already). I am not sure (yet) if a similar issue exist in any other review queue (i.e. the Late Answers queue). What's a concise way to check that environment variables are set in a Unix shell script? I've got a few Unix shell scripts where I need to check that certain environment variables are set before I start doing stuff, so I do this sort of thing:  if [ -z "$STATE" ]; then     echo "Need to set STATE"     exit 1 fi    if [ -z "$DEST" ]; then     echo "Need to set DEST"     exit 1 fi   which is a lot of typing. Is there a more elegant idiom for checking that a set of environment variables is set?  EDIT: I should mention that these variables have no meaningful default value - the script should error out if any are unset. if $(x_0,y_0)$ is local extrema in $ax^2 + by^2 + cxy + dx + ey + l$ then its global too. An exercise on a book asks me to prove that if a point $(x_0,y_0)$ is a local extrema for the function $$f(x,y) = ax^2 + by^2 + cxy + dx + ey + l$$ then it's also a global extrema.  The exercise asks me to consider that $g(t) = f(x_0+ht,y_0+kt)$ is a parabola, but I couldn't understand how it helps. Maybe since it's a parabola, I know that this derivative has to be $0$ in some point, and also that this function $g(t)$ has a global extrema point, but I couldn't derive anything from this. I also tried to find the derivatives, equal them to $0$ and solve the system. I got:  $$x_0 = -\frac{2 b d - c e}{4 a b - c^2}, y_0 = -\frac{   c d - 2 a e}{-4 a b + c^2}$$  But when I try to find   $$f\left(-\frac{2 b d - c e}{4 a b - c^2},-\frac{   c d - 2 a e}{-4 a b + c^2}\right)$$ so I can try to prove:  $$f(x,y)-f(x_0,y_0)$$  is either greater or less $0$ depending on the case (so I can do $f(x,y) \le f(x_0,y_0)$ and $f(x,y) \ge f(x_0,y_0)$)  things get hairy. 'Should've seen it glow' or 'should've seen it glowing'? Which one of the following is the correct one?   I should have seen it glow. I should have seen it glowing.   Or are both correct? Would you parse them please? New SSD problem I've just installed a new SSD replacing an HDD in my old (late 2008) iMac. When I switched it on all I got on screen was a question mark.   On the advice of my son I tried cmdR during switch on with no result.   I then tried alt which at least got me a cursor on the screen, but nothing else. Any suggestions? I've got everything backed up on an external Hard Drive, by the way. SQL Server nvarchar(max) vs nvarchar(n) affects performance Do we really need a [source-code] tag? The  tag doesn't appear to be very useful.  It's used in a variety of ways (for reading, writing, showing, transforming, compiling, translating, obfuscating, obtaining, ... source code) on 1747 questions right now, and doesn't appear to be adding any classification value to the site.  Can we get rid of it?