Proof idea:  $A$ is a skew symmetric matrix.  $A$ is similar to $A^t$ because every matrix is similar to it's transpose. $$A^t = -A $$ $A$ is similar to $-A$. Let $P_{(\lambda)}$ be the characteristic polynomial. $$P_{(A)}=P_{(-A)}=0$$ (I'm not sure about the above step)  And I also know that:  $\overline{P_{(A)}}=\overline{P_{(-A)}}=\overline0=0$  That's where I ran out of ideas. am i in the right direction?
Assume that $A$ is an $n\times n$  real matrix, i.e. $$A^T=-A.$$  Since $\det(A-\lambda I)=\det(A^T-\lambda I)$, $A$ and $A^T$ have the same eigenvalues. On the other hand, $A^T$ and $-A$ also have the same eigenvalues. Thus if $\lambda$ is an eigenvalue of $A$, so is $-\lambda$. If $n$ is odd, $\lambda = 0 $ is an eigenvalue.  A curious search in Google returns  the nonzero eigenvalues of $A$ are all pure imaginary and thus are of the form $iλ_1, −iλ_1, iλ_2, −iλ_2,$ … where each of the $λ_k$ are real.   Here is my question:     How can I prove the fact that "the nonzero eigenvalues of $A$ are all pure imaginary"?
Restore Missing "Settings" (gnome-control-center) Icon in Ubuntu 18.04 I have been running Ubuntu 18.04 for a couple of weeks. I upgraded from 16.04 and all went smoothly.   I am not sure what I changed, but this morning I noticed that the "Settings" Icon that launches gnome-control-center is missing from "Show Applications" and the applications menu as well. When I searched from "Show Applications" for gnome-control-center or just settings, nothing showed up.   So, I figured I must have accidentally removed it and figured a reinstall of the package would resolve the issue. I have searched and found several postings where this worked. I tried:  sudo apt install gnome-control-center   And then rebooted - which reinstalled the gnome-control-center and I could then launch it from the command line. However, there was still no Settings Icon in any of the menus.   I also tried many other suggestions including:  sudo apt-get install --reinstall gnome-control-center sudo apt-get install ubuntu-desktop sudo apt install ubuntu-desktop --reinstall sudo apt-get install unity-control-center   However, none of these has been successful in restoring the icon into "Show Applications" or any applications menu.   Anyone have any suggestions on something I haven't tried? Would be much appreciated.  Or, is there a way to just manually recreate the icon?   Output of locate gnome-control-center.desktop:  /usr/share/applications/gnome-control-center.desktop  /home/dccrens/.local/share/applications/gnome-control-center.desktop /home/dccrens/.gnome/apps/gnome-control-center.desktop   I am running Ubuntu 18.04 desktop on kernel 4.15.0-45-generic. Program to create EMF that conserves transparency What open source tools allow for the creation of EMFs that conserve transparency in ArcMap?  I tried VeryPDF and Any DWG to Image Converter, neither on actually gave me a vector file that provided transparency.  Adobe Illustrator is a bit pricey when all I want to do is convert CAD blocks to vector symbols to use as ArcMap markers.  Anyone know of a good software for this purpose? Finding the nth non square free number We all know what  numbers are. I was more interested in checking an efficient way to find nth non square free number. display only 3 foreach result per row How to apply multiple styles in WPF In WPF, how would I apply multiple styles to a FrameworkElement?  For instance, I have a control which already has a style.  I also have a separate style which I would like to add to it without blowing away the first one.  The styles have different TargetTypes, so I can't just extend one with the other. If Anakin Skywalker didn't get what he wanted as a Sith (to save Padmé), why was he still a Sith? In the prequels, the primary reason Anakin was willing to become  is because he wanted to save Padmé;     ANAKIN: I will do whatever you ask.       PALPATINE: Good.       ANAKIN: Just help me save Padme's life. I can't live without her. If she dies, I don't know what I will do.      PALPATINE: To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret.       ANAKIN kneels before PALPATINE.       ANAKIN: I pledge myself to your teachings. To the ways of the Sith.   Given that he didn't manage to save her, why did he still become a Sith? Here is a fun integral I am trying to evaluate:  $$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$  I thought about integrating by parts $2n$ times and then using the binomial theorem for $\sin(x)$, that is, using $\dfrac{e^{ix}-e^{-ix}}{2i}$ form in the binomial series.  But, I am having a rough time getting it set up correctly. Then, again, there is probably a better approach.   $$\frac{1}{(2n)!}\int_{0}^{\infty}\frac{1}{(2i)^{2n}}\sum_{k=0}^{n}(-1)^{2n+1-k}\binom{2n}{k}\frac{d^{2n}}{dx^{2n}}(e^{i(2k-2n-1)x})\frac{dx}{x^{1-2n}}$$  or something like that. I doubt if that is anywhere close, but is my initial idea of using the binomial series for sin valid or is there a better way?.  Thanks everyone. Just wondering if someone can give me an explanation for this capacitor layout - what is it's function? The chip to the right is an ADC.  Circled in blue.    I'm guessing it is well known but i don't know how to search for it. I have a script which calls two commands:  long_running_command | print_progress   The long_running_command prints progress but I'm unhappy with it. I'm using print_progress to make it nicer (namely, I print the progress in a single line).  The problem: Connection a pipe to stdout also activates a 4K buffer, so the nice print program gets nothing ... nothing ... nothing ... a whole lot ... :)  How can I disable the 4K buffer for the long_running_command (no, I do not have the source)? Show that the following series of function defines a continuous differentiable function function in $\mathbb R$. $$\sum_{n=1}^{\infty}\frac{\sin (nx^2)}{1+n^3}.$$  We have , $|f_n(x)|=\left|\frac{\sin (nx^2)}{1+n^3}\right|\le \frac{1}{1+n^3}\le \frac{1}{n^3}=M_n \text{ (say) }$  As, $\sum M_n$ is convergent so, the given series is uniformly convergent. Also , each $f_n(x)$ is a continuous function in $\mathbb R$. So, the given series converges to a continuous function , say $f(x)$.  But how we can show that $f(x)$ is differentiable function ?  Please help... Is there an IDictionary implementation that, on missing key, returns the default value instead of throwing? I am having trouble with the following problem.      Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(ax-1)$.    I just want a hint.  Basically, I've been looking for a surjective homomorphism from $R[x]$ to $R_D$, but everything I've tried has failed. I think the fact that $f(a)$ is a unit, where $f$ is our mapping, is relevant, but I'm not sure. Thanks How can I prevent inline math formulas from overflowing into the margin? When I compile my document in LaTeX, some formulas in the document in math mode, between $ $, go into the right margin area beyond all the text.  Is there a way to fix this aesthetic problem? How to submit webform programmaticaly? How to remove headers and footers for pages between chapters? I was listening to the song  by Eminem. I've listened it before many times but from the time I've started learning English grammar I am getting confused while reading or listening English. This time I got stuck at the line "Too much pride, between you and I Not a jealous man but females lie". The phrase "between you and I" always sounded correct to me. But recently I learned that me is only used as an object and hearing that phrase sparked my head. Then I searched over the net and found on  that it is actually a hypercorrection.  says,      "...according to many grammarians and stylists a pronoun in a prepositional phrase in English is supposed to be in the oblique case...   ...though there is still disagreement on whether the phrase itself in today's language is grammatically correct or not."   My first question is, why is there a disagreement? Don't we always use the object form in the oblique case? If not then what are those cases?  Secondly, I want to know how common the phrase "between you an I" is? Is it used colloquially by common native English speakers. By common speakers I mean who don't bother grammar. I searched on  and got some results, e.g. . I found the phrase in another song -- . Is it a common mistake among native English speakers or is it used to give some special linguistic affect? I think this phrase has some linguistic affect that is why the singer would have given the name of her song "Between You and I". Because no one would like to use incorrect English in the title of a song, it must have some purpose. Let a be a positive number. Then $\lim_{n \to \infty}[\frac{1}{a+n}+\frac{1}{2a+n}+\cdots +\frac{1}{na+n}]$ Addition is to Integration as Multiplication is to ______ Addition is to Integration as Multiplication is to ______ ? Everyone knows that definite integration  is "a way to sum continuum-many terms" in a rough sense. Can we "multiply continuum-many factors"  in a similar sense? Please Join Us in Welcoming Our New Meta Stack Exchange Moderators! Using ArcMap 10.0, I am working on a map that shows many villages. Each village has a name attribute that I want to use for a label, but in some areas there are so many villages that not all can be labeled. I am using the Maplex label engine and have not checked the "Never Remove (allow overlap)" box.  I am quite happy that not all village names are displayed as this would be way too much. And it is ok that Maplex chooses which labels to display, as I have no way to know which village is more important than another one. This also means that I can't base the labeling on a field such as population or something else, even though that would be less random.  However, I would also like the village points that are not labeled to similarly not be displayed. I think it unlikely that someone would want to see the point without knowing the name of the village. If all points are shown, it is also difficult to tell which point a name label goes with in crowded areas. So instead of choosing just labels not to be shown, I also want the computer to chose features not to be shown.  To give you a better idea on what I mean:    Is there a way to make these features/points not draw if their label is not shown?