With L being a linear operator on a complex vector space $V$ show that $L^{_\dagger}L$ is Hermitian with non negative eigenvalues  I don't really have much of an idea how to do this exercise in such a general example. The proof of non negative eigenvalues is especially eluding me. Any ideas?
Let $L$ be a linear operator on a complex vector space $V$, how to show that eigenvalues of $L^\dagger L$ are positive? I guess this might be a statement taught in a functional analysis class or an advanced linear algebra class?
Which facility upgrade unlocks maneuver nodes in KSP Which  unlocks  in career mode in KSP?  The  hints that maneuver nodes are now unlocked with a facility upgrade:     Are there any perks for upgrading?      The building in question   will upgrade visually, both from the outside and the inside. More   gameplay mechanics will be also be unlocked as the player progresses   through the different building tiers. For example, allowing EVAs in   space or being able to use maneuver nodes in the map view. Why are there so many close votes to review and so few of the others? In the SO Review page , I see the following stats for questions awaiting review:   Close votes:  51.8k (and rising) Suggested edits: 10 (fluctuates, but rarely goes above 20) Everything else: 0 (occasionally see some if you're really lucky)   My question is: Why?  Why are there so many close votes waiting for review? Close votes should expire after four days. And even if there's a good reason for them hanging on longer (ie they're not being viewed), there surely has to be a cut-off eventually after which they can be safely dropped?  And yes, I saw , which is asking basically the same thing. But that was several months ago, and it seems that the queue has hardly moved since then. It seems to me that there's a massive backlog that will never be cleared, and apparently will never expire either. That's not good.  And on the flip side, why are there hardly ever any new user posts or low quality posts to review -- there are always new users posting things, and I recall before this new review system was launched that the list of low quality posts to review was always pretty big. While studying the problem of determining the orbit of a binary star system, my astronomy notes say: "the projection of an ellipse onto a plane is still an ellipse (except the special case in which it's a segment)". How could I prove this?  EDIT: As suggested i'm adding what I know about ellipses. I know the basics of ellipses in the plane, the equation of an ellipse in a plane in cartesia coordinates, the equation of an ellipse in polar coordinates, the fact that ellipses can be obtained by the intersection of planes with a cone. In my geometry course we defined ellipses as the the points of the plane such that the ratio between the distance from a point and a line is constant and less than 1. I also know that the sum of the distances of the points of the ellipse from the foci is constant. Which infinity is meant in limits? For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why? How to unlock device when I forgot my password? I've read  which states that there is an option to disable the lock using my Google account information. However I did not noticed such an option when I entered my password five times wrong. (Although there was also a 20s or 30s delay to complicate brute force attacks.) I use Android 4.0.3 on a Asus Transformer Prime TF201.  Would factory reset work? Benefits of Toad houses I have a few red Toads and a Toad is just hanging out in town. I see an option to buy a red Toad house, ostensibly for him. Are there any added benefits of purchasing this or are the colored Toad houses just cosmetic? How to run grep on a single column? One of my friends argues that princessship is the only word which has 3 identical letter comes together (s) ,but I think there is no word such as princessship.  Can anyone tell me whether this is a real word, and if there are any more English words which have 3 letters consecutively? Looking at the properties for a Windows file I get two attributes, “Size” and “Size on disk,”  and "Size on disk" is always larger.  What do these two metrics mean? How would you evaluate $\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$ How would you evaluate the limit inferior of the sequence $n\,|\mathopen{}\sin n|$? That is, $$\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$$    Edit. Let $\mu$ be the irrationality measure of $\pi$. Since $\mu$ is not known, I will split the question:   Assuming $\mu &gt; 2$, what is $\liminf\limits_{n\to\infty} \ n\,|\mathopen{}\sin n|$? Assuming $\mu = 2$, what is $\liminf\limits_{n\to\infty} \ n\,|\mathopen{}\sin n|$?     I kind of like its graph... How are second order nonlinear ordinary differential equations solved? Is it possible to get Inkscape's bucket fill to entirely fill regions marked by lines? I've noticed there's a slight gap between shapes and lines in which bucket-fill is applied (to the enclosed areas found therein) and the colour fill in Inkscape, is there any way of removing this gap (besides the older-fashion manual way by manipulating the borders of the bucket fill, which is very tedious)? How do I compare strings in Java? Having recently reviewed a few suggested edits, I've found that not being able to see the comments on the post is hindering my understanding of the context of some edits.  In particular, there have been a few posts where someone has made a substantial edit to a question or answer which significantly alters its meaning. Typically I'd reject that as "too radical", since it can't be known that the new meaning of the post was what the original author intended.  However, in some cases it is clear from the comments that the original author is aware of the intended change, and as such the editor is not in fact misrepresenting the author's intentions. In these cases, it would have been useful to see the comments when making the review decision.  As an example, a common cause of this is when an inexperienced user is asked to clarify their question, and ends up writing a comment that would be better off as an edit to the question (due to length, excess code content, etc), presumably due to lack of awareness that the question can be edited. Often, other users will helpfully edit the question to include the information the OP put in the comment (see ,  and  for examples). In this situation, it is clear that the edit reflects the OP's intentions, but this information is only present in the comments which are not visible in the review interface.  Following is the suggested edit screen showing the suggested edit. It seems indentation is increased and one line is moved a few lines upwards and some empty lines added to increase readability. But it look like an invalid edit since changing the related line may cause alteration on the runtime. So not looks lie a valid edit. Better open the answer and examine it        Checking the Q&amp;A itself reveals that a talk had taken place between the OP and the answerer and revision done by the OP with the permission of the answerer. So it was indeed a valid edit.        In short: is there any particular reason comments are not displayed in the suggested edit review interface? I think it would add useful contextual information when making review decisions, and presumably this would be trivial to implement.  Note: this is not a dupe of , as it does not concern comments posted after the suggested edit. Increase word space between words in small caps In polar coordinate how we can get $dx\;dy=r\;dr\;d\theta$?  with these parameters:    $r=\sqrt{x^2+y^2}$ $x=r\cos\theta$ $y=r\sin\theta$  Tanks. How to solve the functional equation $f\left(x^2+f(y)\right)=y+f(x)^2$ How to solve the following functional equation:  Find all $f:\mathbb{R}\to\mathbb{R} $ such that: $$ f\left(x^2+f(y)\right)=y+f(x)^2 $$ Holds for every $x,y\in\mathbb{R}$.  A friend gave it to me, probably its an olympiad question.  I started with the equation $f(f(y))=y+f(0)^2$ which seems to be quite helpful, but I couldn't do it effectively. How to solve it properly? Increasing extent of Raster? Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f. Help me to solve this problem please..  Let $Y_{(1)}, Y_{(2)}, Y_{(3)}, Y_{(4)}, Y_{(5)}$ denote the order statistics of a random sample of size 5 from a distribution having p.d.f. $f(y) = e^{(-y)}, 0 &lt; y &lt; \infty$, zero elsewhere. Show that $Z_1 = Y_{(2)}$ and $Z_2 = Y_{(4)} − Y_{(2)}$  are independent. Hint: First find the joint p.d.f. of $Y_{(2)}$  and  $Y_{(4)}$. Android Preferences: How to load the default values when the user hasn't used the preferences-screen?