Let $f:\mathbb{C}_{4\times4}\longrightarrow\mathbb{C}_{4\times4}$ be the linear map defined by $f(A)=A+3A^T$. Determine eigenvalues, the corresponding eigenspaces and their dimension.  I cannot start this exercise. Some helps?   Thank You
Let $M_n$ denote the vector space of all $n\times n$ real valued matrices and for any matrix $A\in M_n$ let $A^t$ denote its transpose. Define the linear map $T:M_n\rightarrow M_n$ as $$T(A)=A+A^t.$$ What is the minimal polynomial of $T$? What are the eigenvalues and the corresponding eigenspaces of $T$? What is the characteristic polynomial of $T$?  I learned about minimal polynomials of operators from $\textit{Linear Algebra Done Right}$ by Axler. In his description he explains how in order to find the minimal polynomial you must find the smallest positive integer $m$ such that $T^m$ is a linear combination of $I, T, T^2, \ldots, T^{m-1}$. That is, the smallest integer $m$ such that $$T^m = a_0I+a_1T+a_2T^2+ \cdots +a_{m-1}T^{m-1}.$$  As for the eigenvalues...since we are working with real-valued matrices, then $A$ and $A^t$ have the same eigenvalues, so if $\lambda \in \mathbb{R}$ is an eigenvalue of $A$ (and hence of $A^t$) wouldn't $2\lambda$ be an eigenvalue of $A+A^t$? I apologize if I am saying utter nonsense...
Are there any free/open source alternatives to Adobe After Effects? I've been a illustrator and graphic designer for a number of years and now I'm looking to go more into motion graphics and visual effects. Is there anything that is open source/free (not a demo, education version or trial) that is as complex and feature-rich as Adobe After Effects but for Linux? I've tried Cinlerra, kdenlive, openshot and Blender and whilst they are good at video editing (and 3d modelling in Blender's case) they fall short in effects and compositing features.  I've also tried , but the project recently closed and the last update was quite buggy.  Also,  doesn't work and hasn't been updated in many many years, so don't mention it ;-) Dual boot with windows 7! I'm using windows 7 right now. I want to install Ubuntu with dual boot. Is is safe? And does the process go safely? CSS Selector that applies to elements with two classes Is there an official C# guideline for the order of items in terms of class structure?  Does it go:   Public Fields Private Fields Properties Constructors Methods ?   I'm curious if there is a hard and fast rule about the order of items? I'm kind of all over the place. I want to stick with a particular standard so I can do it everywhere.  The real problem is my more complex properties end up looking a lot like methods and they feel out of place at the top before the constructor.  Any tips/suggestions? Limiting range of value types in C++ To prove Cayley-Hamilton theorem, why can't we substitute $A$ for $\lambda$ in $p(\lambda) = \det(\lambda I - A)$? Why doesn't `sudo cd /var/named` work? I have my Xbox One set up to automatically download and install updates, the rest of my settings are shown below:    Is there a reason my games will show an update symbol on their tile, but they still don't download the update until I start the game manually? What could happen if I use 2 cycle fuel in a 4 cycle engine? First off, this is NOT to be confused with the following question:    2 cycle oil is used for my weed eater. There is no way I could use a whole gallon in a year.  Sometimes I run out of gasoline for my 4 cycle lawn mower, and that 2 cycle mix fuel is sitting there getting really old.  If I run that in the lawn mower, am I simply going to create extra pollution or could the 2 cycle fuel cause any damage to the 4 cycle engine? I have a loop going, but there is the possibility for exceptions to be raised inside the loop. This of course would stop my program all together. To prevent that I catch the exceptions and handle them. But then the rest of the iteration runs even though an exception occurred. Is there a keyword to use in my except: clause to just skip the rest of the current iteration? Roots of unity filter, identity involving $\sum_{k \ge 0} \binom{n}{3k}$ How do I see that$$\sum_{k \ge 0} \binom{n}{3k} = (1 + 1)^n + (\omega + 1)^n + (\omega^2 + 1)^n,$$where $\omega = \text{exp}\left({2\over3}\pi i\right)$? What is the underlying intuition behind this equality? Derivative of $(y-XB)' h(y-XB)$ with respect to $B$ How can I perform a t-test (or another hypothesis test) on NPS (Net Promoter Score) results? Here is an explanation of how NPS is calculated:    I'm interested in testing two net promoter scores to determine if they are statistically different. I read a great answer to calculating margin of error for NPS (see link below), but I'm really interested in testing to see if there is a difference between two scores, because I suspect that our results aren't as "different" from year to year as they appear to be.    Is this at all possible? I understand t-tests are typically used to test whether two different record sets are statistically different. But is it possible to test Net Promoter Scores, either with a t-test or some other hypothesis test?  Any ideas you have would be a great help. Thank you! Am I required to pay UK/ USA Visa fees if I have a German residence permit? Here is the situation. I am non-european but staying in Germany with a Residence permit. I can visit all Schengen listed countries with no issue. Assuming that I would want to travel to a non-Schengen country e.g. UK or USA, can I use my residence permit to travel there or am I required to apply as a citizen of my country? Assuming Germans are not required to pay Visa fees to that country, will I have to pay since I only have a residence permit? Third case, Europeans can fly directly to the airport of some countries and apply for a visa from there. In some other countries, you are required to book a Visa appointment at the embassy, which may be granted or not. Does that apply for one with a residence permit? Lagrange Multipler KKT condition I am looking at SVM, and reviewing Lagrange multiplier. Let's say with the constraint function $$g(x,y) = x^2 + y^2 -1 &gt; 0, $$ I am maximizing $$f(x,y) = x + y -1 .$$ Intuitively, since the constraint function does not provide a finite constrained region, there is no solution, and $f(x,y)$ can infinitely increase.  However, if I just plainly proceed with Lagrange multiplier calculation,  $$L(x,\lambda) = (x + y -1) +\lambda(x^2 + y^2 -1)$$ I get the solution: $$y,x = -\sqrt{2}/2, \lambda = 1/\sqrt{2}$$ And this still satisfies KKT conditions although the solution is wrong.  $$g(x) =&gt; 0 , \lambda =&gt;      0, \lambda g(x) = 0.$$  I expected that I would have some result that conflicted KKT conditions, but nothing was violated.   Does this mean that Lagrange multiplier under KKT conditions can produce a solution that cannot be solved, without giving any indication of intractability?  If functions are complicated, what's the systematic way to confirm that the optimization with inequality constraint functions is solvable using Lagrange multiplier? I have the CKeditor plugin installed by itself (i.e. not using WYSIWYG module.) Then, it uses a CKEditor library located in sites/all/libraries/ckeditor. This works fine. I'd like to add more plugins to it; namely, the footnotes plugin. (For the moment, please refrain from "just download a new library from ckeditor.com and replace it"; there were other issues/considerations involved.)  Now, I see this everywhere when I try to look for a solution  config.extraPlugins = 'footnotes';   I tried adding that to admin/config/content/ckeditor/edit// -> Advanced Options -> Custom JavaScript configuration, but it white screens the editor.  I tried adding it to the config.js in sites/all/libraries/ckeditor like below, but it had no effect.  CKEDITOR.editorConfig = function( config ) {   config.extraPlugins = 'footnotes'; };   I can't really add it to a ckeditor.config.js file, dump that file in my theme, and choose "Yes" for "Load ckeditor.config.js from the theme path", since (for content creation/editing) it uses the admin theme which is different from the theme we're developing.  So, I'm at a loss. The only other thing I can think of is to use a hook in the module we have for the site, but I'm not sure if that's write/what hook would be called.  Is it just me, or does it seem like clear documentation on how to manage CKEditor in Drupal is severely lacking? Any help would be appreciated.  UPDATE: After some testing, it looks like the "Custom JavaScript configuration" is supposed to work, but it is possibly broken for CKEditor 4.4+ I read something (can't seem to find it now) that suggested the plugin/widget import was broken on 4.4+, so I changed to 4.3 and the extraPlugins worked. Now, I wasn't able to get footnotes working (which is what kicked this whole thing off), but some other plugins that previously failed during the same steps now work on 4.3. So, there's that. I'm wondering now if it's something with footnotes itself. Resolve build errors due to circular dependency amongst classes Why does paper become translucent when smeared with oil but not (so much) with water? When I smear oil onto a scrap of paper and rub it in, the paper becomes quite translucent; but when I attempt the same with water it doesn't as much. Why? Provide two arguments to one option using getopts In below code when I give option r then getopts requires one arguments:  while getopts ":hr::l:" opt; do     case $opt in         r ) echo "Run Numbers - argument = $OPTARG " ;;         l ) echo "Latency range - argument = $OPTARG" ;;         h ) helptext             graceful_exit ;;         * ) usage             clean_up             exit 1     esac done   But I need to pass two arguments after -r option, instead of one. Is there an easy way to do this? Since Chrome version 38, there's a pretty ugly button in the title bar of the Chrome window that allows you to switch profiles quickly.  Since I'm the only one that uses my computer, this is not really useful to me, and I'd rather reclaim the space for more open tabs.  I've looked in the settings but don't see a way to disable it.  How can I get rid of the button?