Group Orbits of this space under O(3) Consider the vector space: $$ \mathbb{V} = \{ \ A \in M_{3 \times 3}(\mathbb{R}) \ | \ A^{T} = A \mathrm{\ and\ }\mathrm{Tr}(A) = 0 \} $$  As well as the Lie Group: $$ O(3) = \{ \ A \in M_{3 \times 3}(\mathbb{R}) \ | \ A^{T}A = A A^{T} = \mathbb{I} \} $$  Define the group action $\Phi : O(3) \times \mathbb{V} \to \mathbb{V}$ as $\Phi(A,X)= \Phi_{A}(X) = AXA^{T}$. I am to determine the group orbit$s$ of this action.  The way I am understanding this, is that I am to examine the structure of the set $$ S := \{ \ \Phi_{A}(X) \in M_{3\times 3}(\mathbb{R}) \ | \ A \in O(3), X \in \mathbb{V} \ \} $$  I know that symmetric matrices are orthogonally diagonalizable, which means that I can write $X = P D P^{T}$ for any $X \in \mathbb{V}$ (where $P \in O(3)$ and $D$ is diagonal and traceless).  This tells me that $\Phi_{A}(X) = AXA^{T} = APDP^{T}A^{T} = \Phi_{AP}(D)$, where $AP$ is just another matrix in $O(3)$. This tells me: $$ S = \{ \ \Phi_{A}(D) \in M_{3\times 3}(\mathbb{R}) \ | \ A \in O(3), D = \mathrm{diag}(a,b,c) \mathrm{\ with\ }a+b+c=0 \ \} $$  So the orbits of the whole vector space are exactly the orbits of the diagonal matrices in this subspace.  Can I narrow this down further?
Determining the Orbits/Orbit Space of $O(3)$ on Real 3 by 3 Traceless Symmetric Matrices Let $O(n)$ be the set of $3 \times 3$ orthogonal matrices, and $S$ be the set of traceless real symmetric $3 \times 3$ matrices.  Now  $O(n)$ will act on $S$ by conjugation. Note that by the spectral theorem $ \forall A \in S$ in the orbit of $A$ there is a diagonal matrix $D=diag(\lambda_1, \lambda_2,\lambda_3)$ with $\lambda_1\le\lambda_2 \le\lambda_3 $.  Determine the all the orbits of the action &amp; the orbit space of the action.  What I've done:  Since all elements of $S$ are diagonlizable that the orbit of any element should be the orbit of its' diagonalized form because $\forall A \in S,   \exists P \in O(3)$ st  $PAP^t=D $ with $D$ a diagonal matrix and moreover for any other  $Q\in O(3)$ we get $QDQ^t=QPAP^tQ^t=(QP)A(QP)^t=RAR^t$.   And so would that imply the orbit space is just the set of traceless diagonal matrices as $tr(QDQ^t)=tr(DQ^tQ)=tr(Q)$ (cyclic property of trace) and $Q\in S$.  Any help is appreciated.
Apt-get stuck at 0% [Working] Login SSH with .ppk file on Ubuntu Terminal I have a production server. To login to the server I must use a .ppk file.  How to login with Ubuntu Terminal and .ppk file?  I tried this :  ssh -i location/file.ppk username@server-ip   but it is showing an error message. How can I get an underbrace and an overbrace to partially overlap in an equation? Sometimes I want to describe portions of an equation using \underbrace and \overbrace in such a way that the portions are not completely distinct, for example:          /-------\ x = A + B + C + D + E     \-------/   \---/      What is the best way to do this? Intuition for multiple temporal dimensions It’s easy, relatively speaking, to develop an intuition for higher spatial dimensions, usually by induction on familiar lower-dimensional spaces. But I’m having difficulty envisioning a universe with multiple dimensions of time. Even if such a thing may not be real or possible, it seems like a good intellectual exercise. Can anyone offer an illustrative example? How to install Chrome browser properly via command line? I am trying to install chrome browser via command line.  I tried this:  $ apt-cache search chrome browser   The results show that the proper term is "chrome-browser," so I tried that:  $ sudo apt-get install chrome-browser   And then "Y" for the Y/n question.  But the installation threw errors.  Does someone see anything wrong with the commands I issued? In Windows 8 I can reset everything and reinstall Windows. I assume this will clear my drive and start from scratch? Is that right? Does this create a Windows.old folder?  I see this dialog, but aren't my settings and preferences saved in the cloud? If I screw up my settings and reset like this, are my settings also refreshed back to defaults?  Since I use SkyDrive to sync my files, my personal files are safe, right?    Also, in Windows 8 I can refresh my PC without affecting files. Are my desktop and/or Windows store apps uninstalled when I refresh my PC?  I see this dialog, but I still am not sure. For example, when it says apps are "kept" does that mean I don't have to buy them again or are they installed after the refresh is over for me? I assume "from disk" means desktop apps? Or maybe corp apps?    What would motivate a user to choose between these two? I want to have a  video background that will loop on some frames of my presentation. Can anyone help me to do that? Any other words that use “dis-” as an amplifying prefix? I remember hearing once about the etymology of disgruntled, probably based around a joke about how people can not be gruntled. The explanation given was that there was never a word gruntled, rather the dis‑  in disgruntled is actually a usage of the old prefix where it acts like an amplifier.   Firstly, is this true?  Secondly, are there any other words that use this prefix in this way? Maxing out gear with maxed out smithing, enchanting and alchemy I have level 100 smithing, enchanting and alchemy. I also have some nice enchanted gear, fortify smithing and alchemy.   How high should I be able to get my fortified equipment?  Currently I'm around 29% for alchemy and smithing.  How high can I get fortify smithing and fortify enchanting potions without using the console (cheating)? And how? Currently I get +147% and +32% respectively.  Is something like this possible without using the console(or photoshop)? Using a non-centralized structure for tex files - similar to \graphicspath I have recently been thinking about the difference between the two ways of defining an array:   int[] array int array[]   Is there a difference? Why an abstract class implementing an interface can miss the declaration/implementation of one of the interface's methods? Confirming that my own question is a duplicate by voting to close should have immediate effect Someone voted to close my question as a duplicate. I agreed, so I cast my close vote (I have enough reputation to vote to close).  . So I expected this to happen.  But it didn't. This confused me — is this not a question I asked after all? No, it is. Maybe I need to reload the page? Nope, the question still isn't closed. Is the asker confirmation feature broken? Well, there's the “This question may already have an answer here” banner at the top. What if I click “That solved my problem!”? Ah, that does close the question.  It turns out that only clicking “That solved my problem!” instantly closes the question. Casting a close vote has a different effect, it's just an ordinary close vote.  Please make casting a close vote equivalent to clicking “That solved my problem!”. How many nonnegative integer solutions are there to the pair of equations  \begin{align}x_1+x_2+\dots +x_6&amp;=20  \\ x_1+x_2+x_3&amp;=7\end{align}  How do you find  non-negative integer solutions? Why can't I define a static method in a Java interface? Story [novel, novella?] that featured faster-than-light or hyperdrive gained from observing distant drive use, also had a sentient star I have little recollection when I read this, but it started with people observing indications of some special engine use many lightyears away, and it led them to 'reverse-engineer' the traces to create their own drive.  Had the usual tropes of international crew-selection for first-voyage.  Somewhere in the travel, they encounter a sentient star or quasar.  When they get back hundreds/thousands of years later, they arrive to an uncaring Earth that no longer pursues space-travel. How do I load an mp4 video with multiple audio tracks into blender? Propose a privilege to sit in the 500 reputation mark As , the retag privilege serves a lot more confusion and a lot less of its benefit due to the suggested edit system. It must be done away with. It will be done away with. ♪  In light of this decided course of action, we're going to need a new snack for the road between "Create Tags" at 300 reputation and "Established User" at 1000 reputation. We'd like your help in thinking about some ideas on what would be a nice privilege to sit at this level. Since we're... a bit dry on our end.  The primary guideline in deciding what kind of privilege to put here is acknowledging its position in the overall hierarchy of privileges. 500 reputation is an early-intermediate user, one who has started to really get the hang of the site and its function. It can be something that provides convenience in how they currently can act on the site, or it could be some new functionality that fits as an early ability (especially considering there's no new abilities until 2000 reputation's "Edit Everywhat" ability).  I look forward to hearing what creative ideas y'all can provide. Some sort of battle school (no, not Ender's Game), where teens are being trained as military commanders.   The control system for their army hinges around some sort of system that requires extremely fine motor control -- being able to "push" around digital representations of military units using a special interface that interprets minute finger movements.  The boy protagonist eventually meets a girl in the battle school, and is astonished to learn that most people do not live as well as he does -- for instance, he gets a fresh jumpsuit outfit every day and disposes of the old one; this is a luxury.  One scene I recall is them kissing for a moment in some sort of underground passageway; the boy protagonist notes the exact length of the kiss, as he has been trained to have heightened time perception.  I don't recall much about other commander characters; I think they have different special abilities, and one other major character might have physical deformities, possibly because of postwar-radiation-related birth defects.  They have been training for an upcoming, prearranged final battle between all the existing national governments; the military units are very possibly robotic or entirely digital in nature, so as to avoid wasting materiel / human life.    They ultimately win the battle, just barely; I think the girl protagonist gives them an edge in the competition by hacking the timing / control system in favor of their team, causing a discrepancy which only the boy protagonist notices due to his heightened perceptions.  It was agreed that the winning government gets dominion over one of the few islands of land left that was not irradiated in some previous war; the protagonists get to live there as a reward for their victory.  I believe the boy protagonist later realizes he's gradually losing his ability to exactly quantify the passage of time.  Published no later than 2004, likely somewhere between 1980-2000, probably sometime in the '90s. Calculate: $$\int_{0}^{2\pi}e^{R{ {\cos t}}}\cos(R\sin t+3t)\mathrm{d}t$$ My try: $\displaystyle\int_{0}^{2\pi}e^{R{ {\cos t}}}\cos(R\sin t+3t)dt\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\cos(\mathfrak{I\textrm{z}}+3(-i\log z)dz\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\mathfrak{R\textrm{e}}^{(\mathfrak{I\textrm{z}}+3(-i\log z))i}dz\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\mathfrak{R\textrm{e}^{\mathfrak{I\textrm{z}}}}z^{3}dz\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\mathfrak{R\textrm{e}^{\mathfrak{I\textrm{z}}}R}z^{3}dz$ and there is nothing here that is not holomorphic, therefore according to Cauchy theorem it must be exactly $0$.