$A$ is normal if and only if $trace(A^\ast A)=\sum_{i=1}^n|\lambda_i|^2.$ where the $\lambda_i$ are the eigenvalues of $A$.
Show that $A\in\mathbb{C}_n$ is normal $\iff$ $tr(A*A) = \sum_{i = 1}^n|\lambda_i|^2$, where $\lambda_1,...,\lambda_n$ are the eigenvalues of $A$.
How to set AUTO-SCROLLING of JTextArea in Java GUI? Phase Potentials Looking to get a explanation of the following solution.   $x'=sinx$ I understand that you must integrate $sinx$ and then take the negative of it making it $v(x)=cos(x) + c = cos(x)$ with $c=0$. However I am having problems understanding the equilibrium points.   The equilibrium points of this system are,  $$x^*=kz$$ $x*=kz$ k is even, this is unstable $x^*=kz$ k is odd, are stable  Why is it unstable when even and stable when even? How do I deal with a runaway bride? I killed my first wife once I found out there is a mercenary you can marry who gives you up to 300 gold a day (in Whiterun). The only requirement, of course, being that I had to hire her first. So I hired her, and then I popped the question, to which she replied the usual: tell the priest to arrange the wedding.   When we reached Riften I told her to wait, just outside the bee and barb, and arranged the wedding. I went through the process, and the second it finished she left, I followed but as I left she was nowhere to be seen. I have tried going back to my house, the pub where I hired her and that same spot in Riften about 6 times each now, some during the day, some during the night and cannot find her at all.   Now this isn't to much of a problem because I can re-marry...oh wait, to remarry, my spouse has to die, so I can't claim gold off her and can't remarry, and the autosaves have been used up fast travelling to the places I thought she might've been. My last save is something like 6 levels ago, and I don't want to have to load it.   What should I do about this situation? Elementary proof for $\lim\limits_{n \to\infty}\frac{n!e^n}{n^n} = +\infty$ As seen in  question, the series $\sum\limits_{n=1}^{\infty} \dfrac{n!e^n}{n^n}$ diverges. (One way to see this is by noting that the terms of the sum are greater than $1$ and, therefore, don't converge to zero.)  However, more is true: not only are the terms greater than $1$, they blow up to $+\infty$! To see this, one can apply Stirling's approximation and get $\dfrac{n!e^n}{n^n} \sim \sqrt{2\pi n}$.  My question is: is there an elementary proof for $\lim_{n \to\infty}\dfrac{n!e^n}{n^n} = +\infty$?  The reason I'm asking this is that I'm taking a Calculus course and I was assigned the exercise of proving the divergence of the series above and I would feel more satisfied if I could prove that its terms go to $+\infty$ (as opposed to merely proving the terms are greater than $1$.) Obviously, things like Stirling's approximation are outside the scope of the course, so I can't really use them.  (To be clear, you can use anything one learns in $4$ semesters of standard Calculus courses. I hope this restriction is not too obscure.) Why does space expansion not expand matter? Object has no method 'live' - jQuery Can you break up your Attack action for a bonus action? You can do it to move:     If you take an action that includes more than one weapon attack, you can break up your movement even further by moving between those attacks. For example, a fighter who can make two attacks with the Extra Attack feature and who has a speed of 25 feet could move 10 feet, make an attack, move 15 feet, and then attack again. (PHB 190)   It would be necessary for the Great Weapon Master feat:     On your turn, when you score a critical hit with a melee weapon or reduce a creature to 0 hit points with one, you can make one melee weapon attack as a bonus action   As a level 17 Fighter, if I can not place a bonus action between attacks, there are two possibilities:   I have to wait until I finished all 4 attacks of the action, to make another attack as a bonus action the bonus action attack is wasted unless the kill or crit happens on the 4th attack   So can I take a bonus action in the middle of the attacks of an action? Variable with colon dash ${VAR:-} What would you call a person who systematically dumps his work on you in a bid to evade responsibilities (with a work philosophy that when you don't learn or do it, you are never given the responsibilities and thus have no fear of making mistakes) but cluelessly whines to the same people around him for not having all the benefits that they (the higher ranking hardest working colleagues) earn.  PS: Owing to a bad management (Racial discrimination, Poor work ethic,ignorance, etc), if it helps. Proof of property of abelian group Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots. Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.  My attempt:  Consider the polynomial $g=(x-1)(1+x+x^2+x^3+\cdots+x^n)$  As $f\mid g, g$ all the roots of $f$ are roots of $g$.  This means I have to prove the statement:  If $c$ is a multiple root of $f \Longrightarrow c$ is multiple a root of $g$.  Equivalently:  If $c$ is not a multiple root of $g \Longrightarrow c$ is not a multiple root of $f$.  I'll try to prove this last statement:  Suppose $\exists c$ such that $c$ is a multiple root of $g$, then   $$g(c)=0 \implies g'(c)=0$$  $$ g=(x-1)(1+x+\cdots+x^n) \implies  g'=nx^n $$ $$ g'(c)=nc^n=0 \implies c=0 $$ But  $$ g(0)\neq 0 $$ And we've found the contradiction: $\not \exists c: g(c)=g'(c)=0$. Then $f$ has no multiple roots.  Is this correct?   Also: Assuming this is correct, is it possible to prove $\sum_{k=0}^n \frac {x^n}{n!} $ has no multiple roots in a similar manner? Does magic jack work on Ubuntu? Does magic jack work with Ubuntu? I originally had my magic jack on a computer with windows XP and I decided to switch to Ubuntu but I still want Magic jack. how to calculate $$\int_0^\infty \frac{x^{3}}{e^{x}-1} \; dx$$   Be $q:= e^{z}-1 , p:= z^{3}$ , then $e^{z} = 1 $ if $z= 2\pi n i $, so the residue at 0 is : $$\frac{p(z_{0})}{q&#39;(z_{0})} = 2\pi i n ^{3}$$  problem is that this is not symmetric, so how does one find the definite integral? Do the non-zero eigenvalues of AB and BA have the same algebraic multiplicity (for AB and BA not square)? I know that if A and B are square nxn matrices, then AB and BA have the same characteristic polynomial and thus the same eigenvalues (and same algebraïc multiplicity).   I'm wondering though if this can be generalized: if A is a nxm matrix and B a mxn matrix, then AB is a nxn matrix and BA a mxm matrix.  So my question is: will the eigenvalues of AB and BA, that differ from zero, have the same algebraïc multiplicity? Is it safe to use file's hash as IV? I'm encrypting some files using AES in CBC mode.    I'm also using file's digest (SHA-1) to check that data is decrypted correctly (so I need to store it with file).  Is it safe to use this digest as AES's IV and store it in the header of file? Or it has security issue? "as much as you and I" vs. "as much as you and me" Will deleting my question also remove the reputation awarded to answerers? One of my questions got closed, as it was very similar to some other posted questions (though I don't feel the linked questions answered what I was trying to ask).  There's no point to the question staying online, as it doesn't answer anything, so I was going to delete it - but if I do that will it remove the reputation of the helpful answerers I want to upvote? Best way to recover mysql from a "chmod -R 777 /" with databases intact What does an ampersand after this assignment operator mean? Best way to parse RSS/Atom feeds with PHP