Suppose that $A$ is a normal matrix. Prove that one can write: $A = G^4$ for some self-adjoint matrix $G$ Let $A \in  M_n(\mathbb{C})$ be a matrix of order $n \geq  2$. Let $$ p_A(x) = (x - \lambda_1)\cdots(x - \lambda_n) $$ be the characteristic polynomial of $A$ such that all $\lambda_i$ are positive real numbers.  (a) Suppose that $A$ is a normal matrix. Prove that one can write:  (ai) $A = G^4$ for some self-adjoint matrix $G$  Well i can at most prove until the point where there exists an unitary matrix $U$ such that $U^{*}AU = D = G^4$ since $D$'s diagonal contain all its eigenvalues and since they are all real and positive we can split them into $\lambda^{1/4}$ and hence we let $G$ equals to the self adjoint diagonal matrix with diagonals to be $\lambda_i^{1/4}$. HOWEVER, i cannot make it until that $A = G^4$ for some self adjoint $G$ which is very annoying. I mean $A$ and $D$ are similar so im sure we can do it, but i just cant seem to find a way!  Pls help!
Prove normal matrix is 4th power of some self-adjoint matrix Let $A\in\mathbb{M}_n(\mathbb{C})$ be a matrix of order $n\geq 2$. Let $p_A(x) = (x-\lambda_1)\cdots (x-\lambda_n)$ be the characteristic polynomial of $A$ such that all $\lambda_i$ are positive real numbers.  Suppose that $A$ is a normal matrix. Prove that one can write $A=G^4$ for some self-adjoint matrix $G$.  My thinking is that since $A$ is normal, by Principal Axis Theorem, we have $A=UDU^\ast$, where $U$ is a unitary matrix and $D$ a diagonal matrix. However, I fail to see how to proceed. Could anyone help me out?
Calibre No Longer Working Calibre stopped working recently. Tried uninstalling and reinstalling. When I open a terminal and call $calibre I get the following error message: Traceback (most recent call last):   File &quot;/usr/bin/calibre&quot;, line 20, in &lt;module&gt;     sys.exit(calibre())   File &quot;/usr/lib/calibre/calibre/gui_launch.py&quot;, line 73, in calibre     main(args)   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 543, in main     listener = create_listener()   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 514, in create_listener     return Listener(address=gui_socket_address())   File &quot;/usr/lib/calibre/calibre/utils/ipc/server.py&quot;, line 110, in __init__     self._listener._unlink.cancel() AttributeError: 'NoneType' object has no attribute 'cancel'  Do I need to install some dependency that the sudo apt install didn't get? I tried installing via command line and the software installer/updater GUI. I'm on version Ubuntu 20.04.1 LTS. I made sure that my software in general is up to date also. Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2 If low-cost index funds are considered the best investment, why are there so many high-cost, managed funds? The general consensus is that the single best indicator of a fund's future performance is the its expense ratio . If you look for more than 5 minutes online you discover the  approach to investing; the , etc. It looks like the less you pay for a fund (and therefore the less it's managed), the better it does in the long run.  So why is it that in peoples' 401(k)s there are so many high-cost actively managed funds? Furthermore, why are there so many high-cost actively managed funds in the first place? Shouldn't the market reflect the fact that actively-managed funds typically don't beat the index they benchmark to and and react by purging itself of these crappy funds? Given a compact Hausdorff space $X$, if $a \in X$ is a $G_\delta$ singleton (i.e., $\{ a \}$ is a $G_\delta$ set), then there is a countable local base at $a$.   The point $a$ can be written as countable intersection of open sets let's call $U_{n}$.  How can I use the compactness? What is the next step for proof? Could you give me hint please? How does the double strike mechanic interact with other mechanics like lifelink or trample? I understand that the double strike mechanic allows a creature to attack in both a first strike combat damage step and a normal combat damage step.  It's clear that when a creature with no other abilities aside from double strike is blocked that even if the defender has lethal damage marked in the first strike step that the additional damage does not carry forward to the player.  How does this view of double strike change with relation to trample and lifelink?  I believe the correct interpretations are (not accounting for other effects or abilities) that 2 times the attacker's power is marked as damage on the defender.  For trample this amount is reduced by the defender's toughness and additional damage carries forward as normal.  Similarly for lifelink the active player gains life equal to 2 times the attacker's power.  Thus a 10/10 trampling double striking life linker blocked by a 1/1 creature would deal 19 points of trample damage and the active player would gain 20 life.  Is this interpretation correct? running a bash file states:date: command not found I've installed a version of Ubuntu Server and although I'm fairly still fresh with Ubuntu, I'm trying to create a bash file to run a program in the background. More importantly, I'm trying to get a file nameto display correct and create it before I start running the program (I like my debug stuff).  However, here is my bash script:    #!/bin/bash  PATH="/var/log/servers/Game/"  FILENAME="startup"  EXT=".log"  DATE=$(date +%d-%m-%Y)  NEWFILE=${PATH}${FILENAME}_${DATE}${EXT}   I have verified that the file does exist in the PATH directory. However, the issue is with the date command. When I run the script bash -x game.sh this is what is displays.   + PATH=/var/log/servers/Game/  + FILENAME=startup  + EXT=.log  ++ date +%d-%m-%Y  game.sh: line 5: date: command not found  + DATE=  + NEWFILE=/var/log/servers/Game/startup_.log  + echo /var/log/servers/Game/startup_.log   I don't understand why it would say the command doesn't work, when I can go to the main console and run date and get a valid response.  Any help is greatly appreciated! Case insensitive 'Contains(string)' Is there a way to make the following return true?  string title = "ASTRINGTOTEST"; title.Contains("string");   There doesn't seem to be an overload that allows me to set the case sensitivity.. Currently I UPPERCASE them both, but that's just silly (by which I am referring to the  issues that come with up- and down casing).  UPDATE This question is ancient and since then I have realized I asked for a simple answer for a really vast and difficult topic if you care to investigate it fully. For most cases, in mono-lingual, English code bases  answer will suffice. I'm suspecting because most people coming here fall in this category this is the most popular answer.  answer however brings up the inherent problem that we can't compare text case insensitive until we know both texts are the same culture and we know  what that culture is. This is maybe a less popular answer, but I think it is more correct and that's why I marked it as such. Proving an inequality involving the logarithm function: $\frac{1}{n+1} \leq \ln \left(1+\frac 1n\right) \leq \frac 1n$ The question is to prove the inequality $$\frac{1}{n+1} \leq \ln \left(1+\frac 1n\right) \leq \frac 1n\\\forall n \geq 1, n\in \mathbb N$$  I tried using Taylor expansion but couldn't figure out anything. Any ideas? Thanks. Is there a ZIP code I can enter when paying-at-the-pump in the USA with a foreign credit card? In the USA, many  fuel dispensers that accept credit cards require a  that corresponds to the billing address of the credit card.  In Canada, for instance, we have , so it's not obvious what you're supposed to type in since the gas dispensers have only numeric keypads.  Entering '00000' or '90210' seems to always result in failed authorization.  Is there a zip code that works for credit cards with foreign billing addresses? What is ISCAS85 benchmark? What is ISCAS85 benchmark? I find it in a , is there any simulator for it? Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$ Question:  Consider $f: (-\epsilon, \epsilon) \to \mathbb R^{m^2}$ a differentiable path of matrices $m \times m$ such that $f(0) = I_m$ and the function $g: I \to \mathbb R$ is defined by $$g(t) = \det f(t)$$  Show that $g'(0) = \mathrm{tr} \,a $ (trace of the matrix $a$), where $a = f'(0)$.   Attempt:  First we use that $\det f(t) = \det (f_1(t), \ldots, f_m(t)) $, where $f_i$ is the $i$-th row of the matrix $f(t)$ $m \times m$. Considering the operator $': \mathbb R^m \to \mathbb R^m$, $f(t) \mapsto f'(t)$  With this in mind $$\begin{align}g'(t) = \det'f(t) &amp;= \sum_{i=1}^{m} \det (f_1(t),\ldots,f'_i(t),\ldots, f_m(t)) \\&amp;\underbrace{=}_{(*)}\det(f_1(t), \ldots, f_m(t))\, \mathrm tr f'(t) \end{align}$$  For $t = 0$ we have   $$\begin{align}g'(0) &amp;=\det(e_1, \ldots, e_m) \,\mathrm tr f'(0) \\&amp;= 1 \dot\, \mathrm tr \,a \\&amp; = \mathrm tr \, a\end{align}$$  I'm trying to come up with a proof using the fact that $\det$ is a $m$-linear form.   In order to prove $(*)$ I need the following equality, which according to this  holds.   If $A: V \to V$ is a linear operator and $f_0$ is a alternated $m$-linear form, such that $f_0 (e_1, \ldots, e_m) = 1$ then   $$f_0 (x_1, \ldots, x_m) \mathrm tr A = \sum_{i=1}^m f_0(x_1, \ldots, Ax_i,\ldots, x_m) $$  Could someone give a proof to this? How do I decrease the size of the page file? I just upgraded my laptop RAM to 12GB (W510, 4 DIMM slots), and since I am using a SSD, I find hard drive space to be quite precious. However, since I have more than enough RAM for most stuff, I think I can get by with probably around 4GB of page file. however, on C drive, there is a pagefile.sys which I assume to be reserving space for page files in the OS, how would I decrease the size of this file? Shooting many bullets toward the mouse position I try to shoot a bullet toward the mouse position by this code.  float deltaX = mousePosition.x - (aircraft.getPosition().x + aircraft.getLocalBounds().width / 2); float deltaY = mousePosition.y - (aircraft.getPosition().y + aircraft.getLocalBounds().height / 2); float angleRadian = (float)(std::atan2(deltaY, deltaX)); float angleDegree = (float)(angleRadian * 180 / PI);   It works fine with one bullet but when I add another one bullet to shoot together I use this code to specify start shooting point of each bullets.  sf::Vector2f offset1(aircraft.getPosition()); offset1.x += -10;//start shooting point first bullet.  sf::Vector2f offset2(aircraft.getPosition()); offset2.x += 10;//start shooting point second bullet.   this is a picture of this problem.   When I rotate aircraft to picture 2 the bullets are not the same as the first picture.  How can I solve this problem? How do I open the user folder of a hard drive that I transfered? My Windows 10 notebook died but fortunately I was able to rescue the hard drive and put it into another PC.   Unfortunately, the other PC can't open the C/Users/Joe folder. What do I have to do to open the folder? Can't get rid of header X-Powered-By:Express I need a 'good' way to initialize the pseudo-random number generator in C++. I've found  that states:     In order to generate random-like   numbers, srand is usually initialized   to some distinctive value, like those   related with the execution time. For   example, the value returned by the   function time (declared in header   ctime) is different each second, which   is distinctive enough for most   randoming needs.   Unixtime isn't distinctive enough for my application. What's a better way to initialize this? Bonus points if it's portable, but the code will primarily be running on Linux hosts.  I was thinking of doing some pid/unixtime math to get an int, or possibly reading data from /dev/urandom.  Thanks!  EDIT  Yes, I am actually starting my application multiple times a second and I've run into collisions. How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$ How would one prove that $$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$  where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio? Any canon Shadowrun material related to NY/PA/NJ areas? I'm looking for specific, canon (i.e. official, published) references to material for the Shadowrun game-world setting. My game setting is based around the Philadelphia area; I've turned New Jersey into a radioactive swamp, then dropped a nuke in it; I've got some things established for the Philly area, and NYC.  What I want to know is whether anyone knows of specific references to published material dealing with this region. Any mention of the big apple in Shadowrun's history? Any famous names in the area, or megacorps that are mentioned? Anything at all that's part of the published SR work, that I can incorporate, or at least avoid contradicting.  If you have any home-brewed material for that area, I would love to see a link in comments, but please don't put it as an answer. Thanks! =) Ruby -- capitalize first letter of every sentence in a paragraph What is so special about Prime?