If A² = I, prove that the matrix A is diagonalizable.  I have computed the eigenvalues to be 1 or -1 but I'm not sure how to proceed from here.   I'm thinking along the lines of "since rank(A + I) + rank(A - I) = n, therefore there exists n linearly independent vectors which corresponds to n eigenvectors. Hence, A is diagonalizable". Is that correct?  Thanks in advance!
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?
Recovering an accidentally deleted  folder in OSX - no Time Machine or backup available I am a longtime Windows user, recently switched to Mac. I replaced folder with another folder (they had the same name) thinking that OS X will merge the contents of both folders. But OS X deleted the original folder and replaced it with the new one. Windows merges both the folders (I do this often in Windows). I have now lost hundreds of files. I haven't enabled Time Machine. I've tried using Finder's Undo, but that doesn't work.  Currently, I am trying to recover my data using . It shows estimated time as 7 hours, 31 minutes. Does anyone know any free data recovery software that's faster and more powerful? Why is dark matter the best theory available to explain missing mass problems? Why is dark matter the best theory to explain the missing mass problem?    Why is dark matter mathematically necessary to explain the missing mass problem?    On a side not I believe dark matter is definately intuitive the next logical step in physics is to accept the fact the direct observations can not always be the determining factor in the acceptance of a theory.  Positivism is not going to be the only driving force in physics forevermore.    One theory that seems almost as good is the supersymmetric particle. What determination which are actually detectible and how would we distinguish the differences between the two?    References:   . How to concatenate characters in java? How do you concatenate characters in java? Concatenating strings would only require a + between the strings, but concatenating chars using + will change the value of the char into ascii and hence giving a numerical output. I want to do System.out.println(char1+char2+char3... and create a String word like this.  I could do   System.out.print(char1); System.out.print(char2); System.out.print(char3);   But, this will only get me the characters in 1 line. I need it as a string. Any help would be appreciated.  Thanks collapse vs minimize How to specify the SMTP server in PHPMailer? To start in or at a company? I am not sure about what is right:     I can start my career at an international company.   or     I can start my career in an international company.   For me at sounds appropriate? Question: Let $F_n$ the sequence of Fibonacci numbers, given by $F_0 = 0, F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Show for $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$   My (very limited) attempt so far: after creating a small list of the values $F_0=0, F_1=1, F_2=1, F_3=2, F_4=3, F_5=5, F_6=8, F_7=13, F_8=21, F_9=34, F_{10}=55$ i can see that yes it does seem to work for instance $F_{6+3}=F_5 F_3 +F_6 F_4 = 10 +24 = 34 = F_9$. However, I really don't know where to begin as showing that this must hold in general terms. Should I be looking to use limits? Or perhaps induction? What is the best way to solve this? Why copy constructor not invoked here What is the difference between UNION and UNION ALL? What is the difference between UNION and UNION ALL? Custom Cursor Image CSS I have a number of heroshot images, that have a modal popup when clicked.  I'm trying to get my cursor to turn into  magnifying glass whenever it is moved over the image. The following CSS does not appear to work even though my magnify.cur is present in the right location.  a.heroshot img {   cursor:url(/img/magnify.cur), pointer; }   Has anyone ever done anything similar? I don't mind a JavaScript solution if one exists.  EDIT: It works in Safari, but not in Firefox. The difficulty is that it should be cross platform. Windows 2000, XP, Vista, OSX, Linux, other unix variants. I am looking for a snippet of code that can accomplish this for all platforms, and a way to detect the platform.  Now, you should be aware of  that user.home does not work correctly, so please do not provide me of textbook answers, I can find these myself in the manuals. Chinese Passport Holder Transit in South Korea A Chinese passport holder with a US visa has a flight scheduled from Beijing to Los Angeles in mid-November.  This person wants to go to South Korea for a 3 day conference; the conference is one week prior to the PEK-LAX flight.  If this person buys a round-trip ticket from China to Korea, will the transit visa be sufficient? Or will they need a regular visa?    Extra info:  What if the original ticket were modified to make it ICN-PEK-LAX?  . It says "Holding a confirmed onward flight ticket for departure within 30 days." Would a return ticket to China than a flight to LAX a week later be sufficiently "onward"? Is there a PPA with the latest version of LibreOffice? There was recently another release of , but it is  a pain to have to manually go to their website and download the packages every time there is a new release. Is there a PPA that always has the latest version of LibreOffice? Constants of integration in integration by parts Example of a finitely generated faithful torsion module over a commutative ring Can a finitely generated module $M$ over a commutative ring have $\operatorname{Ann}(x) \neq 0$ for all $x \in M$ while $\operatorname{Ann}(M) = 0$?  It's not difficult to show that there is no such module if the ring is a integral domain. For general, I guess the answer is yes. But I failed to find a desired example. Within, between or overall R-square for random effects in Stata How to detect character sets encoding for all of language? I have the following code that will remove lines with the pattern banana and 2 lines after it:  sed '/banana/I,+2 d' file   So far, so good! But I need it to remove 2 lines before banana, but I can't get it with a “minus sign” or whatever (similar to what grep -v -B2 banana file should do but doesn't):  teresaejunior@localhost ~ &gt; LC_ALL=C sed '-2,/banana/I d' file sed: invalid option -- '2' teresaejunior@localhost ~ &gt; LC_ALL=C sed '/banana/I,-2 d' file sed: -e expression #1, char 16: unexpected `,' teresaejunior@localhost ~ &gt; LC_ALL=C sed '/banana/I,2- d' file sed: -e expression #1, char 17: unknown command: `-' Stack Exchange app iPhone X optimization The Stack Exchange app doesn’t look very good on Apple’s brand new flagship phone yet.    Also the app can’t identify the device model either. (Says Unknown (iPhone 10,3))   App Version: 1.6.6.2 Device: iPhone X (iPhone10,3) OS Version: Version 11.1 (Build 15B93) Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$ $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$, $|z|&gt;1$.  There are two facts that my professor uses that I am confused about.  The first is: $|1+z^n| \geq ||z|^n-1|$, I believe this is true for any $|z|$.  The other is: $\frac{1}{|z|^n-1} \leq \frac{2}{|z|^n}$, I believe this is also true for any $|z|$.  Can anyone prove these statements for me?