I am trying to find the eigen values of a skew symmetric matrix. So let $A$ be a skew symmetric matrix so $A^T$ = -$A$ Let $\lambda$ be  an eigen value of $A$. So there exists a non-zero $x$ such that $Ax$ = $\lambda$$x$ So $(Ax)^T$ = ($\lambda$$x$)$^T$ or $x^TA^T$ = $\lambda$$x$$^T$ or $x^TA^T$$x$ = $\lambda$$x$$^T$$x$ or $x^T(-A)$$x$ = $\lambda$$x$$^T$$x$ or $-x^T\lambda x$ = $\lambda$$x$$^Tx$ or $\lambda$ =  -$\lambda$ (since $x$ is non-zero) or $\lambda$ = $0$ I am not able to understand what is wrong in this. Help, please!
Given a real, skew-symmetric matrix $\mathbf{A}\in\mathbb{R}^{m\times m}$, and a nonzero vector $X\in\mathbb{R}^{m}$, classify the scalar $\lambda$ (real, complex, imaginary, etc.) in the eigenvalue equation $$ \mathbf{A}X = \lambda X $$  The matrix $\mathbf{A}$ is skew symmetric if $\mathbf{A}^{T}=-\mathbf{A}$.  My try : I am new to matrices , so can not get a idea how to deal with it , Any hint will help
Can you help me with my software licensing issue? This is a  about Licensing.    Questions on licensing are off-topic on Server Fault.  If your question has been closed as a duplicate of this question, then this is because we want to help you understand why licence questions are off topic rather than just telling you "it just is". In all likelihood, this question will not address your question directly, it was not meant to.    I have a question regarding software licensing. Can the Server Fault community please help with the following:   How many licenses do I need? Is this licensing configuration valid? What CALs do I need to be properly licensed? Can I run this product in a virtual environment? Can I downgrade this product to an earlier version? Am I entitled to feature X with license Y? $f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$.  How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ?  I am just trying to prove the convergence, not that it converges to $l$ (that part be deduced immediately by Stolz-Cesaro and the sequential characterization of limits) I would like a way to mark a question in the Unanswered "my tags" page as something I'm not interested in or able to answer The /unanswered/tagged?tab=mytags page gives a list of questions from my favorite tags that are currently considered "unanswered" - they have no up voted or accepted answers. I think it's good to put them in front of me easily, but even though they have one or more tags that I'm interested in, I may not have the knowledge, ability, or interest to write an answer to them. However, they will remain on this page until someone writes an answer.  I would like to be able to dismiss questions from views, like this one, that show me interesting questions. I may choose to share them on my social media to get them views and answers, but I don't want to continually see them. Instead, I want to be able to use views like unanswered/tagged?tab=mytags as a queue of things that could use my attention. If I run along the aisle of a bus traveling at (almost) the speed of light, can I travel faster than the speed of light? If $\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1$, then what's the value of $4\left(\frac{\cos^3y}{\cos x}+\frac{\sin^3y}{\sin x}\right)$? Is there any decent speech recognition software for Linux? The short version of the question: I am looking for a speech recognition software that runs on Linux and has decent accuracy and usability. Any license and price is fine. It should not be restricted to voice commands, as I want to be able to dictate text.  More details: I have unsatisfyingly tried the following:       (e.g., )  (used to run  on Linux but was discontinued years ago)      (built on the Kaldi speech recognition toolkit)    +  +  +   : only accepts voice commands  All the above-mentioned native Linux solutions have both poor accuracy and usability (or some don't allow free-text dictation but only voice commands). By poor accuracy, I mean an accuracy significantly below the one the speech recognition software I mentioned below for other platforms have. As for Wine + Dragon NaturallySpeaking, in my experience it keeps crashing, and I don't seem to be the only one to have such issues unfortunately. On Microsoft Windows I use Dragon NaturallySpeaking, on Apple Mac OS X I use Apple Dictation and DragonDictate, on Android I use Google speech recognition, and on iOS I use the built-in Apple speech recognition. Baidu Research released  the  for its speech recognition library using  implemented with Torch. Benchmarks from  are encouraging as shown in the table below, but I am not aware of any good wrapper around to make it usable without quite some coding (and a large training data set):      System Clean (94) Noisy (82) Combined (176)     Apple Dictation 14.24 43.76 26.73   Bing Speech 11.73 36.12 22.05   Google API 6.64 30.47 16.72   wit.ai 7.94 35.06 19.41   Deep Speech 6.56 19.06 11.85     Table 4: Results (%WER) for 3 systems evaluated on the original audio. All systems are scored only on the utterances with predictions given by all systems. The number in the parentheses next to each dataset, e.g. Clean (94), is the number of utterances scored.  There exist some very alpha open-source projects:   (part of Mozilla's Vaani project:   ())  Vox, a system to control a Linux system using Dragon NaturallySpeaking:  +     (to be released by Google, mentioned at Interspeech 2018)  I am also aware of this  as well as this .  I am aware of  , which allows speech recognition via Dragonfly on one computer to send events to another, but it has some latency cost:  I am also aware of these two talks exploring Linux option for speech recognition:   (David Williams-King)  (Tavis Rudd) When and why did Harry Potter fall in love with Ginny Weasley? In the earlier HP books it's pretty clear that Harry and Ginny get on quite well as friends (and that she's rather smitten with him) but I'm struggling to identify when and why Harry started to like Ginny in a romantic fashion..  I feel like it is simply presumed that Harry suddenly likes Ginny Weasley back, but do the books actually describe when and why he started to like her that way?  No speculation, please. Canon answers only. I have heard that there are some Pokémon that are region specific.  Could someone please clarify which Pokémon are region specific and in which regions that they can be found in? Number of suggested edits seems to include own edits I am attempting to solve this counting problem through combinatorial argument. The following is the equation I am given:   $$\sum_{i=1}^n (i-1)(n-i) = \binom{n}{3}$$  I understand that the right-hand side of this equation represents a set of $n$-elements out of which we choose 3. For example I believe we can say suppose we have a group of $n$ people and we want to choose 3 out of $n$ to be in a committee. However I'm not sure how to express the left-hand side in words. If forming a committee is an appropriate way to tackle this problem then I know the left-hand side must utilize the addition and multiplication principles, but I don't know how to put it into words. Also my intuition tells me that in solving this we should first flip $$(n-i)(i-1)$$  Thanks! "Knowledge of" vs. "knowledge on" I have a sheet in which I first apply a filter to a certain column to show "Unique records only". Now I have a column which contains, for e.g., values like:  Applepie Applesauce Crabapple Banana Mango   What I need is the count of all visible cells containing the word "Apple". Note that this should exclude the hidden (filtered out) cells. Now I have found out a formula:  `=SUMPRODUCT(SUBTOTAL(3,OFFSET(Sheet1!A:A,ROW(Sheet1!A:A)-MIN(ROW(Sheet1!A:A)),,1))*(Sheet1!A:A="&lt;Text to search&gt;"))`   However the problem with this one is this formula searches for the entire text. NOT just a part of it. How do I modify this formula to return count of cells containing "Apple" i.e. for this example it should return 3 as the result. Two question on harmonic function Could someone please help me with this proof?     Suppose that $a, b \in N$, and $d = \gcd(a, b)$. Since $d$ divides $a$, we have $a = de$ for some integer $e,$ and similarly $b = df$ for some integer $f$. Prove that $\gcd(e, f) = 1$.   I understand why it works. Since d is all the common factors of $a$ and $b, e$ and $f$ had no common factors, therefore the $\gcd(e,f) = 1$. But how do I prove this?  Thanks in advance. bash aliases equivalent for powershell? By default my Windows PowerShell starts in C:\Users\Santosh, my XAMPP installation is in D:\ so the htdocs folder is located at D:\xampp\htdocs. If I have to edit something in htdocs folder then I have to type full cd D:\xampp\htdocs\ (autocompletion is not so kind) then edit that file.  If this PowerShell were a Bash I would do this in .bash_aliases file:  alias htdocs='cd D:\xampp\htdocs'  Is it possible to maintain Bash aliases like file and alias any command in PowerShell? Please add the ability to fold blocks of code in questions and answers It would be great to have the option to fold code snippets, entire questions and answer posts on a user basis and allow defining some generic foldable regions within the question/answer editor.  Non generic folding info could be stored in cookies perhaps to move the storage/processing requirement to the client side.  Definable folding can take any free and safe HTML tag or other multi character trick (&lt;>Fold this&lt;>) How should I act when I see tiny "Have you tried this..?" type of answers? Is there a name for "the string of characters typed randomly on a keyboard home row"? Execute vs Read bit. How do directory permissions in Linux work? If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$?