for Spin 1/2, we have Pauli matrix as in . So what's the generalized 3-by-3 Pauli matrix for spin 1 or even larger spin? Is there a generalization method?
What is the exact $SU(2)$ representation to which these Hermitian generators belong? \begin{equation} t_a=\{t_1,t_2,t_3\}=\left\{\frac{1}{\sqrt{2}}\begin{pmatrix}  0 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 1 \\ 0 &amp; 1 &amp; 0 \end{pmatrix},\,\frac{1}{\sqrt{2}}\begin{pmatrix}  0 &amp; -i &amp; 0 \\ i &amp; 0  &amp; -i \\ 0 &amp; i &amp; 0 \end{pmatrix},\,\begin{pmatrix}  1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; -1  \end{pmatrix} \right\} \end{equation}  I'm a bit puzzled with this, these are not the generators of the triplet representation of $SU(2)$ (the triplet in $SU(2)$ is the adjoint rep which is real and constructed with the structure constants), nevertheless they are used as if they were in much literature. What are really these generators? They seem like the extension to 3 dimensions of the Pauli matrices. The 3 dimensional fundamental rep (this does not make sense to me)? some kind of non-irreducible representation?
Looking for a book series about a girl and her friends captured and sold as slaves Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$  My attempt: I took A =  $$         \begin{bmatrix}         2 &amp; 0  &amp;0\\         0 &amp; 0  &amp;2\\         \end{bmatrix} $$  B= $$         \begin{bmatrix}         1 &amp; 0  \\         0 &amp; 0  \\         0 &amp; 1  \\         \end{bmatrix} $$  It satisfies given condition and I get $\det(BA)=0$   But I have not proved it  How do I prove that it is always zero (background)I am 12th grader and I know about adjoint,inverse,determinant,rank of a matrix and the other basics.  However I do NOT know about eigenvalues and eigenvectors. Get total amount of time between records when changed state I have a table, showing the status (healthy, broken) for each device at a given timestamp. I need to get the total amount of time of a whole breakdown.  So the data is stored like this:  device_owner   device_id    timestamp          status owner1         device_1     2001-01-01 09:00   0 owner1         device_2     2001-01-01 09:15   0 owner1         device_1     2001-01-01 09:30   1 owner1         device_2     2001-01-01 09:45   1   and so on.  The example above shows    device_1 was down from 09:00 to 09:30 (total 30 mins),   device_2 was down from 09:15 to 09:45 (total 30 mins)   What I need  Need to calculate the total amount of time for the whole breakdown for owner_1 will be 45 mins (from 09:00 to 09:45) instead of 60 mins (30 + 30).  device_owner   total_breakdown_min owner_1        45   What I've done  Currently I can define a breakdown time for each device separately (device_1 = 30mins, device_2 = 30mins). For this I transformed the table using LEAD over partition by device_owner, device_id and date part from timestamp:  device_owner  device_id     timestamp        status lead_timestamp   lead_status owner1        device_1     2001-01-01 09:00   0     2001-01-01 09:30     1 owner1        device_1     2001-01-01 09:30   1     NULL                 NULL owner1        device_2     2001-01-01 09:15   0     2001-01-01 09:45     1 owner1        device_2     2001-01-01 09:45   1     NULL                 NULL   So the datetime difference between timestamp and lead_timestamp in sum gives me the total breakdown time for a device.  ;WITH LeadStatus AS (     SELECT         D.*,         lead_status = LEAD(D.status) OVER (PARTITION BY D.device_owner, D.device_id, D.date ORDER BY D.datetime ASC),         lead_timestamp = LEAD(D.datetime) OVER (PARTITION BY D.device_owner, D.device_id, D.date ORDER BY D.datetime ASC)     FROM         #DeviceStatus AS D ) SELECT * FROM LeadStatus   It is the same device list for any owner. The data snapshot is to be taken between 2 dates (StartDate, EndDate). The records are written to a DB each n seconds and also when the device status changes. I need this desperately but can't find it. Am I doing something wrong? Lebesgue measurable but not Borel measurable The new profile shouldn't mistakenly imply that you can earn the Informed badge on a meta site I just joined Sports, and I decided to take a look on Sports Meta. I clicked on my profile there. This is part of what I saw:     Yet the profile shouldn't indicate that it is, in theory. . .  This seems to be a bug. Text formatting now allowed in comments (List of proven and disproven abilities) On Takodana,      the main characters and various bar patrons can see the planet Hosnian Prime being destroyed by Starkiller Base with their (respective) naked eyes. How did they see this? Is the Hosnian system close enough to see? If it's not very close and they could just see the explosion, how did the light from it reach them so quickly? Evaluation of $\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}$ How do I change the GRUB boot order? Uneven line height in XeTeX I'm embarking on a journey to create a conference abstract book using XeTeX and got stuck within five minutes:        The missing 'g' in 'somethin' influences the line height of the following line. How can I correct this?  At the end I want to be able to include the abstracts and corresponding information through our online registration system, so this should be a global setting.  My MWE:  \documentclass[10pt, a5paper]{article} \usepackage{fontspec}  %% GEOMETRY \usepackage[left=2cm, right=1.5cm]{geometry} % to change the page dimensions  \newenvironment{AbsHead} {     \fontsize{16}{22}\headfont\noindent }% {     \vspace{0.5 cm} }  \newenvironment{AbsAuthors} {     \fontsize{12}{14}\authorfont\noindent }% {    \vspace{0.5 cm} }  \begin{document}  ￼￼￼￼\setmainfont{Minion Pro} \newfontfamily\authorfont{Helvetica Neue Light} \newfontfamily\headfont{Helvetica Neue Condensed Bold}  \begin{AbsHead}The XIAP  something something something somethin somethin somethin somethin somethin somethin somethin something something\end{AbsHead}   \begin{AbsAuthors}Somebody Someone\end{AbsAuthors}   \noindent Purpose: Lorem ipsum \end{document} I want to learn Linux Kernel and I've downloaded the latest kernel version. Now I want to configure Linux with make menuconfig and I'm receiving an error:     Arch/x86/Makefile:184  Compiler lacks asm-goto support...   Can somebody help me? Can you find all classes in a package using reflection? Is it possible to find all classes or interfaces in a given package? (Quickly looking at e.g. , it would seem like no.) Who took care of Naruto when he was young? In episode 350, we see that the Third Hokage took on to take care of Naruto. But towards the beginning of the anime Naruto is seen to be living alone.   Did Naruto live with the Third? If so, why did he leave there? Is there any way to increase the drop rate of items? changed primary email in google account. how to log in? I added gmail to my google account. Now when i try to log into stack exchange using google it prompts me to create a new account instead of using the existing.   The google account is still the same but google changed the primary email adress.  How can i log in again? On , there are download options for only certain logos/icons. Shouldn't this include all the graduated sites logos/icons? I think it should. Feature request - links underlined CMake & CTest : make test doesn't build tests Starting scheduled task by detecting connection of USB device I know it has been discussed that it is not possible to start an application from an usb drive upon connection due to limitations of autorun (or autoplay??) in Win 7. But it is possible to create a scheduled task having an event type trigger. Surely there must be an event occurring when the drive -- or any USB device, for that matter -- is connected.    Does anybody have the slightest idea which Event ID should I use? Or at least what type of event? Where can I find the event in event viewer?