How to apply Givens rotation to a matrix with complex elements? I am trying to apply  to a matrix, I could do it when the matrix had real values. For example, for this matrix: $A=\begin{bmatrix}3&amp;5\\4&amp;1\end{bmatrix}$ I can simply calculate $sin$ and $cos$ like this: $r=\sqrt{3^2+4^2}=5$ $cos=3/r=0.6, sin=4/r=0.8$ But when the values are complex numbers, I have a problem in determining which value of $r$ to select. For example: $A=\begin{bmatrix}3+1.5i&amp;5-7i\\4-2i&amp;1+3i\end{bmatrix}$ $r=((3+1.5i)^2+(4-2i)^2)^{0.5}=4.403-0.795i$ or $-4.403+0.795i$ So the question is, Which value of $r$ should I select? What is the criteria?
QR factorization of complex matrix If you have two complex numbers $a,b$ how can you find the QR factorization of   $ M = \begin{bmatrix}        aI_n\\          bI_n \end{bmatrix} $,  I can't seem to be able to do it. I tried an earlier trick, but I now know it's not necessary true for complex numbers. Is there anyway to do this?  I think if I use a Givens algorithm I should be able to reduce it to a QR factorization.
How to fully delete a git repository created with init? In the definition of , I read the following text:     Many of the act's provisions were to sunset beginning December 31, 2005, approximately 4 years after its passage. In the months preceding the sunset date, supporters of the act pushed to make its sunsetting provisions permanent, while critics sought to revise various sections to enhance civil liberty protections. In July 2005, the U.S. Senate passed a reauthorization bill with substantial changes to several sections of the act, while the House reauthorization bill kept most of the act's original language. The two bills were then reconciled in a conference committee that was criticized by Senators from both the Republican and Democratic parties for ignoring civil liberty concerns.   What does sunset mean? In which cases can the word be used with a similar meaning? Let $d=(a^n,b^n), d_1=(a,b)^n$; need show $d\mid d_1$ &amp; $d_1\mid d$ to prove.  For the general case of $(a,b)\ne 1, (a,b)= (d_1)^\frac{1}{n}$, else $(a,b)=d_1$.  Let, $d_2= (a,b)=d_1^\frac{1}{n}\implies a=a'd_2, b = b'd_2\implies a^n= (a')^nd_2^n, b^n= (b')^nd_2^n$  Unable to progress further. Intuition behind why order of a subgroup divides order of a group. Suppose $f$ is continuous on $[a,b]$ and differentiable on (a,b). Does it follow that $f'$ is continuous on $(a,b)$? Difference between pts and tty 'An avalanche of mail'? - it should be 'mails', shouldn't it? I came across this example on WordWeb while going through the meaning of the word ''.     the program brought an avalanche of mail (Just like "a group of man men").   I'm quite sure it should be mails. Kindly confirm. Let $|A|=\alpha, |B|=\beta, |C|= \gamma$ be cardinals and $\beta \leq \gamma$. Prove $\alpha ^{\beta}\le \alpha ^{\gamma}$.  So from the given we know that there's an injection $f:B\to C$ and some functions $h:B\to A, g: C\to A$. We want to prove there's an injection $l_1:A\to C$. It appears that $f$ doesn't help here.  Trying to take representatives from $A$ and show they're in $C$ and there's an injection doesn't work so maybe the function should be $l_2: h \to g$ but I don't know how to work with it. Install a single specific package from future distribution I'm currently using precise. I need to upgrade a certain package (namely  and it's dependencies) to a newer version, which is currently found in raring (alfa at the time of writing this). How to do it ?  What will happen if I just add to my sources.list:  deb http://archive.ubuntu.com/ubuntu raring main   Will the next apt-get update / apt-get upgrade, upgrade most of my system to raring then ? Outlining in Blender? I am currently working on converting my logo vectored as a SVG to a 3D model using the 2.72 version. I ran into a problem with my logo because it contains a black stroke and an orange stroke (see both images below), but currently all I have is the interior color of each shape. I was wondering whether or not there was a simple way to add in these strokes, and how exactly to go about creating the strokes. Thank you in advance community! How to remove the whitespace BEFORE itemize/enumerate? I'm somewhat of a LaTeX noob, and this is driving me completely nuts.  When I type  \documentclass{article} \usepackage{enumitem}  \begin{document}     Some text     \begin{itemize}[noitemsep]         \item Item 1         \item Item 2         \item Item 3     \end{itemize} \end{document}   I don't want the space which appears in between the text "Some text" and the itemization to be there:    When I search around on Google all I see is tons of long, complicated advice on this site, StackOverflow, forums, etc. on how to not fix the problem ("parskip is evil", etc.), or tons of vague/confusing answers on what package to look at, with no actual example for me to learn from.  So while this question is already asked many times, I have never found a single concrete answer to this question.    The question couldn't be simpler: How do I "properly" get rid of this extra space? Set of discontinuity of monotone function is countable The result I am trying to prove is the following:     Let $F:\Bbb R\to\Bbb R$ be increasing. Then the set of points, at which it is discontinuous, is countable.   I have been reading this form Folland's Real Analysis and he proves this by considering the sum $\sum_{|x|&lt;N}[F(x^+)-F(x^-)]$ which has to be finite by some sort of telescoping sum. Now, I am not sure how he defines this sum. I am assuming he does this by using nets (let me know if there is another way to look at this), but he doesn't introduce the concept of nets until the next chapter. So I was looking for some other ways to prove this result and I found the following result (from Stein-Shakarchi) which is quite similar but assumes $F$ to be bounded.      A bounded increasing function $F$ on $[a,b]$ has at most countably many discontinuities.   This is shown by using the density of rationals in $\Bbb R$, i.e. $\exists r_x\in\Bbb Q$ s. t. $F(x^-)&lt;r_x&lt;F(x^+)$ corresponding to each point of discontinuity $x$.   Now I feel like there shouldn't be a problem in doing the exact same thing for unbounded increasing function. Is this right? Secondly how does this proof relate to the first one? I feel like there isn't much of a difference between the two proofs. To be precise I think that the previous sum is well defined/finite because of this countability of discontinuous points. What do you think about this way of thinking?  Edit. Here is the Folland's proof in more detail:  Since $F$ is increasing, the intervals $(F(x^-),F(x^+))$ are disjoint for all $\forall x\in\Bbb R$. Moreover,  $\forall|x| &lt; N$ such intervals are contained in $(F(-N),F(N^+))$ and so  $$\sum_{|x|&lt;N}[F(x^+)-F(x^-)] \leqslant F(N) - F(-N) &lt; \infty. $$ Hence the set $\{ x \in (-N,N) : F(x^+) \neq F(x^-)\}$ is countable. Does the universe have a center? If the big bang was the birth of everything, and the big bang was an event in the sense that it had a location and a time (time 0), wouldn't that mean that our universe has a center?  Where was the big bang? What is there now? Are we moving away from the center still? Are these even valid questions? How does the Ranger's Extra Attack feature work with Two-Weapon Fighting? I think I know the answer to this one but I'm double checking.  At the 5th level the Ranger gets Extra Attack, which lets them attack a second time during the Attack action. If they are fighting with 2 weapons, the second weapon normally uses their bonus action.  Does this mean when using the Extra Attack and using two weapons, a ranger can attack twice during the Attack action, and still make 1 extra attack as a bonus action, for a total of 3 attacks? "Images with street signs" CAPTCHA impossible to solve Faculty positions: What to say in a statement on diversity? How to center text vertically within a textbox in Photoshop Difference between Ethernet splitter and switch I have one Ethernet port that is wired directly to the router on another level.  However, I want to run a desktop and a server both off this one Ethernet port.  Can anyone tell me the difference between an Ethernet splitter and a switch?    Also, will either the splitter or switch slow down the connection as opposed to just one connection? On the importance of the i.i.d. assumption in statistical learning Did Snape ever commit murder? It's well known that Severus Snape was, in one way or another, a Death Eater since he was at Hogwarts. He hung out with the likes of Avery and Mulciber who became well known Death Eaters, he joined Voldemort right after he left Hogwarts, and he basically became Voldemort's right hand man until the very end.   However, in his time as a Death Eater did he ever kill anyone? We know that he was a Death Eater for around 3 years before Voldemort fell; Lily and James died at 21 and they were the same age as Snape, so it would have been 3 years since they left Hogwarts. Then after Voldemort came back in book 4, Snape would have been a Death Eater again for around 3 years. That's a total of around 6 years of Death Eaterhood with Voldemort.  So, in either of these periods, did Snape commit murder as part of his job as Death Eater? I'm not counting a certain death at the end of book 6 for reasons which are obvious if you've read book 7.   Clearly it's much more likely he killed before Voldemort fell since that roughly correlated with the period when he pledged allegiance only to Voldemort, before he became a teacher at Hogwarts. But did he ever do it?