Find spectrum for matrix $A$ Let $A = \left[ \begin{array}{*{20}{c}} 0&amp;b&amp;0&amp;0&amp;0&amp;0\\ c&amp;0&amp;b&amp;0&amp;0&amp;0\\ 0&amp;c&amp;0&amp;b&amp;0&amp;0\\ 0&amp;0&amp;c&amp;0&amp;b&amp;0\\ 0&amp;0&amp;0&amp;c&amp;0&amp;b\\ 0&amp;0&amp;0&amp;0&amp;c&amp;0 \end{array} \right]_{n \times n}$.  Why does matrix $A$ have spectrum $\sigma (A) = \left\{ 2\sqrt {bc} \cos \left(\frac{\pi k}{n + 1}\right):k = 1,2,\ldots,n \right\}$?
find spectrum matrix A Let $A = \left[ \begin{array}{*{20}{c}} 0&amp;b&amp;0&amp;0&amp;0&amp;0\\ c&amp;0&amp;b&amp;0&amp;0&amp;0\\ 0&amp;c&amp;0&amp;b&amp;0&amp;0\\ 0&amp;0&amp;c&amp;0&amp;b&amp;0\\ 0&amp;0&amp;0&amp;c&amp;0&amp;b\\ 0&amp;0&amp;0&amp;0&amp;c&amp;0 \end{array} \right]_{n \times n}$.  Why does matrix $A$ have spectrum $\sigma (A) = \left\{ 2\sqrt {bc} \cos \left(\frac{\pi k}{n + 1}\right):k = 1,2,\ldots,n \right\}$?
Rules for Profiles Are there any rules concerning what I can put in my profile? Story ID: A pre-teen girl who is a genius crosses the country in a van Why does $1+2+3+\cdots = -\frac{1}{12}$? $\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) &gt; 1$.  Why should analytically continuing to $\zeta(-1)$ give the right answer? What happens if I use RAM of a faster speed than the CPU supports? Looking at the  CPU, under Memory types it says "DDR3-1333/1600". My question is, in what way does the CPU limit the RAM speed? What would happen if I put in a DDR3-2133 RAM chip together with the CPU, if the motherboard supports the faster RAM speed? Would it be better to get a RAM chip that is DDR3-1600 so it matches the CPU's listed frequency? How to install Python 3.6? Why have header files and .cpp files? Why does C++ have header files and .cpp files? Diagonalizable matrix with parameters $$     \begin{bmatrix}     -2 &amp; -2 &amp; 0 &amp; 0 \\     -b &amp; -b &amp; 0 &amp; 0 \\      0 &amp; 0 &amp; -4 &amp; 4 \\      0 &amp; 0 &amp; b &amp; -b \\     \end{bmatrix} $$  I have this matrix and I need to find the values for $b$ that this matrix won't  be diagonalizable.  For what I got this matrix is diagonalizable for every $b$ value.  Please write me also the final answer, thank you very much. Iptables rules with ! character I have entered iptables rules to log a specific connection on our server, but it doesn't seem to be working as I want it to. Here are the rules:  1    LOG        tcp  -- !10.51.0.0/16         0.0.0.0/0           state NEW    tcp dpt:8040 LOG flags 0 level 4 prefix `New Connection_8040TCP: ' 2    LOG        tcp  -- !192.168.0.0/16       0.0.0.0/0           state NEW tcp dpt:8040 LOG flags 0 level 4 prefix `New Connection_8040TCP: '   Iptables should log only IP connections which are not in the specified networks, but it keeps logging all connections from all IP addresses. How to compare objects by multiple fields Assume you have some objects which have several fields they can be compared by:  public class Person {      private String firstName;     private String lastName;     private String age;      /* Constructors */      /* Methods */  }   So in this example, when you ask if:  a.compareTo(b) &gt; 0   you might be asking if a's last name comes before b's, or if a is older than b, etc...  What is the cleanest way to enable multiple comparison between these kinds of objects without adding unnecessary clutter or overhead?   java.lang.Comparable interface allows comparison by one field only Adding numerous compare methods (i.e. compareByFirstName(), compareByAge(), etc...) is cluttered in my opinion.   So what is the best way to go about this? Why don't we read ''k'' here?  Know → /nəʊ/   Knotty → /ˈnɒt.i/   Knockout → /ˈnɒk.aʊt/ Process management and high technology weapon. Novel name and author? How to explicitly discard an out argument? Error in if/while (condition) {: missing Value where TRUE/FALSE needed Prove that if $\lim_{n \to \infty}z_{n}=A$ then:  $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n})$$ where $N$ is value of $n$ for which $|A-z_{n}|&lt;\epsilon$ then taking the limit of this sum devided by $n$ , and noting that the second sum is as close as you wish to $nA$ while the first is as close as you wish to $0$. Not sure if this helps.... Meaning of hash symbol in pin name I'm reading the datasheet of a PSRAM (available ). In the top-right hand corner all the pins are listed. Some of the pin names end with a hash, e.g. LB#, OE#.  What does the # notation mean for pins? Your UK visa application has been issued.      PLEASE DO NOT GO TO THE VISA APPLICATION CENTRE BEFORE THEY CONTACT YOU.      You will be contacted by the Visa Application Centre when your documents are ready for collection, or have been dispatched via courier where this is relevant.       You are reminded that you should avoid making travel plans until you have collected your passport, as the outcome of your visa application is not final until you receive your passport.      Please ensure you check the details on your visa when collecting your passport. If you need to contact us you can do so via our International Enquiry Service at www.gov.uk/contact-ukvi-outside-uk      Kind regards,      UK Visas and Immigration   British Embassy Abu Dhabi     Your UK visa application has been issued.   I know words like processed, forwarded, received, refused but never heard anything like that, so I am confused. Small snap loop devices visible in gnome-disk-utility OR what is the function of snap ubuntu-core I just realized using gnome-disk-utility (3.18.3.1) that my 16.04 installation is featuring several small loop devices.  In detail:   78 MB Loop Device (/var/lib/snapd/sn_ntu-core_352.snap -> /dev/loop0) 76 MB Loop Device (/var/lib/snapd/sn_ntu-core_216.snap -> /dev/loop1) 79 MB Loop Device (/var/lib/snapd/sn_ntu-core_423.snap -> /dev/loop2) 705 KB Loop Device (/var/lib/snapd/snaps/tpad_12.snap -> /dev/loop3) 684 KB Loop Device (/var/lib/snapd/snaps/tpad_11.snap -> /dev/loop5) 705 KB Loop Device (/var/lib/snapd/snaps/tpad_13.snap -> /dev/loop6)   Image showing gnome-disk-utility    My questions are as follows:   What are those loop devices used for? Is that an expected behaviour in 16.04? Can  I remove them? And if so - how (just by unmounting)? Is there any risk in removing the 2 installed snaps mentioned below (tpad and ubuntu-core)   UPDATE  Installed snaps  According to snap list I do have   tpad (1.8.1) ubuntu-core (16.04.1)   installed. I remember I did tinker with snaps for a short time when  I realized this option - but am pretty unsure if  I  installed those 2 snaps - or if they are pre-/auto-installed.  tpad is a Terminal text editor with GUI-like user interface  I  did install manually, as  I don't use it  I removed it now via sudo snap remove tpad. As a result the 2 loop devices pointing to tpad are gone in gnome-disk-utility.  So - at the current point  I am still having 1 snap installed called ubuntu-core and I don't know what it is used for. The summary of this particular snap is   The ubuntu-core OS snap ［I'm Done］ button is available after ［Save Edits］ even when nothing was edited in First Posts queue Can I write the output the of time difference command to a file? Areas versus volumes of revolution: why does the area require approximation by a cone? Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$.  Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder.  However, if we want to find the surface area, then we approximate it as part of a cone and the formula is $$\int_a^b 2\pi f(x)\sqrt{1+f'(x)^2} dx.$$  But if approximated it by a circle with thickness $dx$ we would get $$\int_a^b 2\pi f(x) dx.$$  So my question is how come for volume we can make the cruder approximation of a disk but for surface area we can't.