How do you change something into a matrix? E.g)      How does $2+3i$ become $\left(\matrix{2 &amp; -3\\3 &amp; 2}\right)$   Is  there a specific rule for changing an input into a matrix that has to be learned? Or how is it done?  Thank you.
Possible Duplicate:          On Wikipedia, it says that:     Matrix representation of complex numbers   Complex numbers $z=a+ib$ can also be represented by $2\times2$ matrices that have the following form: $$\pmatrix{a&amp;-b\\b&amp;a}$$   I don't understand why they can be represented by these matrices or where these matrices come from.
Restricing domains for universal and existential quantifiers I am currently using Rosen's "Discrete Mathematics and Its Applications" (7th ed.) for my discrete mathematics course. We recently talked about quantifiers, more specifically the universal and existential quantifier. What confuses me is when we started talking about restricting the domain (i.e. instead of all real numbers, only all real positive numbers), where does the conditional statement come from when restricting the domain of a universal quantifier and where does the conjunction come from when restricting the domain of a existential quantifier?  This link did help:   However, the part that confuses here is how are the domains restricted in the answer given in the link above? The text for  says:     {1}{G}, {T} : Target creature you control fights another target creature. (Each deals damage equal to its power to the other.)   Does this have anything to do with combat? Or is it just direct damage between two creatures? For example, can one of the creatures have defender? Or what if one of them is tapped?   I've been assuming it is more like the damage dealt by a .  Also: are the two creatures then “tapped” until their next untap step? ie: could you use Ulenvald Tracker to make two creatures fight in your pre-combat step, then enter the combat phase with a creature who just “fought” untapped? Could you make two creatures fight who are both tapped? Using the same secret key for encryption and authentication in a Encrypt-then-MAC scheme Is it a weakness to use a single shared secret for protecting messages using a Encrypt-then-MAC scheme?    Assuming a system is using AES-256-CBC and a SHA1-HMAC and the same secret key for both operations.  Upon intercepting one of these messages and attacker could brute force this secret key by trying to solve the HMAC or by successfully decrypting the first block of the ciphertext.  Performing  a decryption on a single block of AES-256 should be a lighter operation than performing a SHA1-HMAC on a potentially large message.  Therefore AES-256 is ideal for a massively parallel brute-force attack and the use of the SHA1-HMAC doesn't hurt the system.  Is this approach sound, or is there an attack that has not been taken into consideration? What is the equivalent of Persian idiom "When the reed blooms"? In Persian, we say "When the  blooms" when we want to  express that an event:   Never happens. (This is only the opinion of the speaker so it's not a fact) It's very unlikely to happen. It's going to take so long to actually happen. (It's so long that you'd be better off giving up expecting it)  3.1. We also use it when someone is always postponing something they have to do so you never actually see it happen.   For example:      Kyle: When are you going to pay my money back?      Cartman: When the  blooms.   Now I don't know if I got the plant's name right because there are some other versions like:        In any event, they have mentioned this plant in the idiom because apparently some versions of it actually never bloom. Others believe it's because farmers harvest them before they get the chance to bloom (e.g to make music instruments), so it never happens. I'd love to know the equivalent idioms or expressions for this.  PS: I only know "When pigs fly" but I'm not a fan of it because the word pig sounds a bit rude to me and partly because it implies a strong impossibility which doesn't work for the #3 use. App keeps crashing & rendering error since update to 1.0.68 On my mac I have 2 accounts and the both are normal account. So in this moment I don't have a Administrator account, only one that was in the past a administrator, but now is not.  How I can have again a Administrator account? No more updates for Google Chrome 32-bit Difference between  and Highlight \verb in inline text I am trying to check if the user_picture field is empty using the twig template, but I am not able to do this.  I tried lots of ways:-  {% if node.user_picture.value %} {% if node.user_picture.entity %} {% if content.user_picture.value %} {% if content.user_picture|render.entity %} {% if content.user_picture|render %}     Sample of code:    {% if node.user_picture.value %}     &lt;p&gt;User-picture 02&lt;/p&gt;     {{ user_picture }}   {% else %}     &lt;p&gt;Anymonus-picture 02&lt;/p&gt;     &lt;div&gt;       &lt;a href="#"&gt;&lt;img src="{{base_path}}/WEBSITE/themes/THEME/images/anonymous.jpg" alt=""&gt;&lt;/a&gt;     &lt;/div&gt;   {% endif %} SQLiteException while inserting values in the database in Android Java Reflection Performance Does creating an object using reflection rather than calling the class constructor result in any significant performance differences? I'm fairly new to LaTeX and I want to know how to change the way sections are numbered by default. I use LyX, and there it is relatively simple to change the way an enumeration is numbered, i.e.  1 foo  a bar    i baz   But I want to know if there is a way to do this by default for sections and subsections. I have now enough reputation points on Stack Overflow to edit every post directly. But sometimes (especially if this is about a field where I'm not savvy, or when I try to edit a question title without having understood the question itself) I might be not totally sure about whether my edit really is an improvement.  For now, I have these options:   Do the edit anyway, hope that someone else will catch it if it is wrong Don't do the edit, and hope that someone else will see this post and edit it in the right way Let a comment and hope that someone will see it (either after editing or instead of it). Flag the post for moderator attention (but will the moderators know it better?) Search the right chat room to find someone who knows more about the topic.   All these are not really optimal, I think.  I would like to have a check box     [_] Submit this edit for approval   which, when checked, would submit the edit into the usual edit approval queue. Then I would get at least one or two other pairs of eyes looking over my edit, and maybe even improving it.  (I don't really care about the +2 reputation for the edit - I think it should be without any reputation difference here.)  There's an  for nearly giving everyone the ability to suggest edits that the post owner can approve, but I'm simply interested in being able to insert them into the general edit queue where anyone can approve them.  Actually, I could do this right now by having a low-reputation sock puppet (or a totally anonymous edit), but I think this is not the intended use for sock puppets, is it? Also, I would like to be able to get feedback for my edit. How can I disable the purple splash screen at boot? How can I disable the purple "Ubuntu" splash screen that displays at boot before the login screen has loaded? I'd prefer to just see plain text scroll by. Defaulting to the sign-up form? Recently I've noticed more and more sign-in / sign-up workflows defaulting to the sign-up option first and providing a minimal "Sign-In / Login" action somewhere else on the page. What's the reason behind this practice?  Examples where I've seen this recently: Is $f(x,y)=(xy)^{2/3}$ differentiable at $(0,0)$? I looked at the graph of $f(x,y)=(xy)^{2/3}$ at the origin, and no matter how far I zoom in it has four corners along $x=\pm y$, so it doesn't look like it is differentiable as $z=0$ doesn't look like a tangent plane.  I also found that its partial derivatives are not continuous at $(0,0)$, e.g.  $$ \lim_{(ky^2,y) \to (0,0)} \frac{\partial f}{\partial x} = \lim_{(ky^2,y) \to (0,0)} \frac{2}{3}x^{-\frac{1}{3}}y^{\frac{2}{3}} = \frac{2}{3}k^{-\frac{1}{3}} $$ (although $\frac{\partial f}{\partial x}(0,0) = \frac{\partial f}{\partial y}(0,0) = 0$)  I also tried to prove/disprove its differentiability by definition, and for it to be differentiable I need $$ \lim_{(x,y) \to (0,0)} \frac{f(x,y)-0}{\sqrt{x^2+y^2}} =0 $$ but then I got stuck. Am I even on the right track?  [UPDATE] I need to do the same thing for $f(x,y)=(xy)^{\frac{1}{3}}$... I think the answer is that $(xy)^{2/3}$ is differentiable but $(xy)^{1/3}$ is not, according to my professor but I'm not sure Enlarge a partition without losing data with Windows 7 diskmgmt? I have a 320GB hard drive currently partitioned as below:     A long time ago, a computer expert formatted my hard drive like this because I wanted to install Ubuntu. I'm now ready to do this, but I'll never need so much free space (103.09GB). I want to enlarge the D:\ volume because it already has lots of data on it.  I know that when I open diskmgmt.msc, I could possibly enlarge the D:\ volume using some free space from the unformatted 103.09GB volume, because the program offers me the option to enlarge. However I'm afraid that I might lose data doing this.  Is it possible to enlarge the partition with this Windows 7 tool or am I better using GParted via "Try Ubuntu" on my Live CD? Would either of them cause data loss in this scenario? How does Java calculate negative numbers? What if residuals are normally distributed, but y is not?