Number of Jordan boxes of size j for eigenvalue $\lambda$ is $2\dim\ker(A-\lambda I)^j-\dim\ker(A-\lambda I)^{j+1}-\dim \ker(A-\lambda I)^{j-1}$
Jordan form, number of blocks.
Was Luke aware the Emperor was a Sith Lord? Similar to , I'm asking the more specific question, was Luke aware that the Emperor was a Sith Lord? I don't recall it being mentioned in any of the films.  If yes, at what point does he learn this? If not why did Yoda or Obi-Wan not warn him?  Perhaps he could have avoided being defenseless and struck by Force Lightning if he'd been made aware? Let $A$ be a square matrix of order $n$ such that $A^2 = I$.  Prove that if $1$ is the only eigenvalue of $A$, then $A = I$. Prove that $A$ is diagonalizable.   For  (1), I know that there are two eigenvalues which are $1$ and $-1$, how do I go about proving what the question asks me? Is "chutzpah" used by non-Jewish English speakers? I came across this statement while studying electric currents and I am confused:  "There is no electric field inside a conductor. Hence no current can flow through it".  Is there a fallacy in this statement?  Any help would be appreciated.  The conductor being discussed is ideal. What is the driving force for the current in this case? Draw a vertical line over the entries of a column in an array Windows 8 Pro clean install - everything slow; hard drive at constantly at 100% Yesterday, I have downloaded and installed Windows 8 Professional, after downloading it (legally) and burning it to a DVD, on my 3-year-old  (Intel Core Duo 2x2.4&nbsp;GHz, 4&nbsp;GB RAM,  hard disk drive).   The installation failed two or three times, each time differently (error codes, for example 0xc000021a, with an infinite boot loop), but finally I completely formatted the hard drive and installed. Seemingly, everything was OK.  However, after booting successfully, the OS is extremely slow. When I open the task manager, it shows that the HDD is at 100% - however, no application or background process or Windows process can be made responsible. I searched the Internet and found that Windows might try to "index" the hard drive, but I did't find a process that isn't also on my second computer, which still uses the Release Preview (restarting every two hours). Also, the startup is empty, and HD Tune and HDD Health show that the drive health is OK, even though I don't know how reliable these two programs are.  I looked for diagnose software from Toshiba, but they only offer tools for "Fujitsu branded Toshiba devices"; I tried it out, and it did not work on my hard drive.  Does this sound like a hard drive failure? The laptop is 3.5-years-old, has been carried around sometimes roughly and was used (and on) most of the time. However, it would have failed exactly the night before installing Windows&nbsp;8 Profesional, which seems - unlikely.  If this is a software error - then what did I do wrong if this occurs from the moment of installation (even the personalization at the end of installation took half an hour)? How can I fix this? I'm struggling with the following question;     For every graph $G$ such that $K_3 \not\subseteq G$ (i.e. $G$ does not contain a triangle), prove that $\chi(G) \leq 2\sqrt{n} +1$ (where $\chi(G)$ denotes the chromatic number).   Any hints? I really cannot see the connection between the triangle-free nature of $G$ and a chromatic number being bounded above by $O(\sqrt{n})$. Let $f(x)$ be a monotone non-increasing function such that $\int_{0}^{\infty}f(x) dx$ converges. Prove: $\lim\limits_{x \to \infty} xf(x)=0$. My question is, why can't I simply contradict any other possibility by using the Integral Limit Comparison Test with $1\over x$? I am, after all, to show that $f(x)=o(x)$ as $x\to \infty$. I really don't understand why monotony is crucial here. I could use some help. Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$ I need help with a textbook exercise (Stein's Complex Analysis, Chapter 3, Exercises 9). This exercise requires me to show that $$\int_0^1 \log(\sin \pi x)dx=-\log2$$ A hint is given as "Use the contour shown in Figure 9."    Since this is an exercise from Chapter 3, I think I should use the residue formula or something like that. But the function $f(x)=\log(\sin \pi x)$ becomes singular on $x=0$ and $x=1$, which makes the contour illegal for the residue theorem. Can anyone give me a further hint on this problem? Many thanks in advance!  P.S. This is my first time on Math Stack Exchange. If you find my post ambiguous, let me know. Does there exist a function such that $\lim_{x \to a} f(x) = L$ for all $a \in \mathbb R$ but $f(x)$ is never $L$? How do i install drush for a drupal site which is on shared hosting? The shellscript is very simple:  retroarch $* -c /tmp/retroarch/Data/retroarch/gambatte.cfg   ("retroarch" is another shellscript)  If the filename passed contains spaces it fails:  RetroArch [ERROR] :: Could not read ROM file. Error in updating to ADT rev 23 for Eclipse Prevent questions with movie/tv shows/game spoilers from appearing in the hot questions list I've noticed that occasionally "hot questions" from SciFi and the Movies &amp; TV network contain major spoilers about the movie/show discussed.  The Matrix spoilers:      (just now there's something about Cypher betraying Morpheus in the Matrix, which is a big deal for people who haven't seen the movie).   I like browsing through those but fear that one day one of these questions will spoil a show I'm excited about and yet to finish   Breaking Bad spoilers:      (like the finale of Breaking Bad, imagine a question asking why did one character betray another at the end or something).   Can users posting such questions be asked to label those with a tag (such as 'spoilers') and then somehow have the network hide those from the "hot questions" list? Or possibly format them differently?  Similarly, this could apply to the Arcade network too.  P.S. Love how this question spikes in popularity around the time GoT starts running again. I swear it should be fairly simple technically to ban questions that have a tag that says 'spoilers' from appearing in the Hot Network Questions.  P.P.S. Oh, and with every fresh release of Star Wars (The Last Jedi being the most recent one)!  P.P.P.S. Another year, another Game of Thrones, another spike in popularity! How to convert a factor to integer\numeric without loss of information? Electric circuits in TeX, LaTeX, and Friends What are some ways to draw electric circuits in TeX systems?  I'm making this community wiki since I haven't found a question related to this.  I was thinking along the lines of . What do you think of that? Other examples are more than welcome, of course. What are "Community Wiki" posts? Some questions and answers are marked Community Wiki and are owned by a community wiki user:  (In fact, this very question is one of them)  Why have Community Wiki posts? How do Community Wiki posts work? How does a post become a Community Wiki post? How can the Community Wiki status be removed from a post? How can I find Community Wiki posts? Wacom Tablet - Illustrator brush tool I currently got the intuos art wacom tablet so still learning how to use it. I'm currently in illustrator trying to draw out a map and the brush tool is doing something weird and I cannot figure out why. You can see in my image that where there are curves the black slightly spills over and is filling in half the curve with black. Brush setting is set up as a new calligraphic brush and the size is set to pressure, 3pt-3pt. Configure proxy for APT? Let $f:R\rightarrow R$ be a function such that all its successive derivatives exist in all $R$ and also $f(x)f''(x)\leq 0$ everywhere.  If $\alpha$ and $\beta$ be two successive roots of $f(x)=0$.   Then prove that $f'''(x)=0$ for atleast one $\gamma \in(\alpha,\beta)$.  My Attempt:  I began by taking an example $f(x)=(x-1)(2-x)$ and let $x=1$ and $x=2$ be its two successive roots.The statement is trivially true.  Then I took the function $f(x)=e^{-x}(x-1)(2-x)$.The statement is true here also so on and so forth.  But what would be the exact proof I wonder. It appears to be a question of Rolle's Theorem.