I was studying block matrices and suddenly this question came to my mind. Let $A, B \in \Bbb R^{n \times n}$. From this , $$\det \begin{pmatrix} A &amp; B\\ B &amp; A\end{pmatrix} = \det(A-B)\det(A+B)$$ even if $A$ and $B$ do not commute. Does a similar condition hold for the following block matrix? $$\begin{pmatrix} A &amp; -B\\  B &amp; A \end{pmatrix}$$
The complex general linear group is a subgroup of the group of real matrices of twice the dimension and with positive determinant.  Let us decompose complex matrices $M$ as $M=A+iB$, where $A,B$ are real matrices. Now consider the correspondence $$f(A+iB)=\begin{pmatrix} A &amp; -B \\ B &amp; A\end{pmatrix}.$$  If $\det f(M)=|\det M|^2$ for square matrices, then we would have $GL(n,\mathbb C)\subseteq GL_+(2n,\mathbb R)$ with the identification $M\to f(M)$, which is an injective homomorphism. In other words, the complex general linear group would be a subgroup of the group of real matrices of twice the dimension and with positive determinant.  How is $\det f(M)=|\det M|^2$?
How do I run a command? I am a new user, who has just been told to run / execute a command. The answer has some text with a grey background like this text here or  sometimes this text here.   However, I don't know what "run" or "command" means in relation to Ubuntu.  How do I do this?    This is intended for use as a Q&amp;A you can link to when you tell someone to run a command. How to edit 'etc/hosts' file in non-rooted phone? Where to ask "subjective and argumentative" questions on programming? Javascript file uploads , clicking &quot;link&quot; &quot;share&quot; for anwers yields a short (traceable) URL in the &quot;share a link to this answer&quot; dialog, such as †:  These short URLs show no clue about the linked question whatsoever. Also, when different people use a short URL for the very same post, then the user id suffix makes those be different URLs for a browser. Such links will then not be shown as visited‡. I know how to get URLs that have a proper slug. But as I see  pop up on these sites (where referrals ), I assume many others just don't realize full URLs would be better for use on these sites themselves? Or many don't want to do the little effort to get the full URL? So: can we please have an easy way to get the full URL as well, in a way that is easily found for those who don't know about the different links? (Currently for questions, clicking &quot;link&quot; has the same effect. But then one can also right-click the title, or take the URL from the location bar, to get the full human-readable URL. Once, Jeff .)  † The short links for answers are not (yet) [automatically converted][6] into a clickable question title either, unlike links such as `https://meta.stackexchange.com/questions/74274/privacy-leak-in-permalink#75963`, which renders as https://meta.stackexchange.com/questions/74274/privacy-leak-in-permalink#75963 But I guess that will change soon.  ‡ On Meta the CSS for posts doesn't indicate such difference anyway, which is a different issue. Let $\lambda(A)$ be the Lebesgue measure of $A$. There exists an interval $I$ such that $\lambda(E \cap I) / \lambda(I) < 1-\epsilon$ (Not mentioned in title but $\epsilon$ is a number greater than $0$ and $E$ a Lebesgue measurable subset of $\mathbb{R}^n$.) I know a question equivalent to this one has been asked (). But it was not answered. Also my thinking is a bit different. So I would love to get some help on this problem. So far I've got:  $$\frac{\lambda(E \cap I)}{\lambda(I)} &gt; 1- \epsilon \iff \frac{\lambda(I)-\lambda(E\cap I)}{\lambda(I)} &lt; \epsilon$$  And also, a theorem states that, for all $\epsilon$ there exists a closed set $F\subset E$ such that $\lambda(E\setminus F) &lt; \epsilon$ so it would sufice to prove  that there exists an interval $I$ such that $$\frac{\lambda(I)-\lambda(E\cap I)}{\lambda(I)}&lt;\lambda(E\setminus F).$$ I'm not sure if this helps me much but any help is appreciated. What would you do? How do you reset timers? When to use 'by' instead of 'at' or 'on' to specify time/date? I had a problem with disappearing marker as the data sum up to more than 100 while xmax was set to 100. See .  As written in the answer to the cited question I tried to avoid the problem by normalizing the value to 100. But this lead to (much smaller) rounding errors in previously correct rows and now the marker disappear there. See the following example. Before the normalisation the column F ist affected, after the column B.   How can I in a reliable way avoid this rounding problem? Or at least get a clear error message. I have quite a number of this plots and I don't want to have to check them all the time for missing markers.    \documentclass{scrreprt} \usepackage{pgfplots} \pgfplotsset{compat=1.13} \usepackage{pgfplotstable} \pgfplotsset{  my stackbar plot/.style={              xbar stacked,              xmin=0,xmax=100,              symbolic y coords={A,B,C,D,E,F,G},              ytick=data,              nodes near coords={xxx},}}  \begin{document}  \pgfplotstableread[col sep=space]{  text   --  -    +       ++  A      0.0 1.7 13.8    84.5  B      0.0 0.6 20.1    79.3  C      0.0 1.9 13.2    84.9  D      0.0 1.6 27.9    70.5  E      1.3 3.9 19.5    75.3  F      0.0 1.4 15.0    83.7  G      0.3 1.7 24.8    73.2  }\data   \pgfplotstablecreatecol[create col/expr={\thisrow{--}+\thisrow{-}+\thisrow{+}+\thisrow{++}}]{sum}\data %  \pgfplotstablecreatecol[create col/copy=--]{--o}\data  \pgfplotstablecreatecol[create col/copy=-]{-o}\data  \pgfplotstablecreatecol[create col/copy=+]{+o}\data  \pgfplotstablecreatecol[create col/copy=++]{++o}\data  \pgfplotstablecreatecol[create col/expr={100/\thisrow{sum}*\thisrow{--o}}]{--}\data  \pgfplotstablecreatecol[create col/expr={100/\thisrow{sum}*\thisrow{-o}}]{-}\data  \pgfplotstablecreatecol[create col/expr={100/\thisrow{sum}*\thisrow{+o}}]{+}\data  \pgfplotstablecreatecol[create col/expr={100/\thisrow{sum}*\thisrow{++o}}]{++}\data  \pgfplotstablecreatecol[create col/expr={\thisrow{--}+\thisrow{-}+\thisrow{+}+\thisrow{++}}]{sumnew}\data  \pgfplotstabletypeset[columns={text,sum}   ,precision=10,columns/text/.style={string type}]\data \quad \pgfplotstabletypeset[columns={text,sumnew},precision=10,columns/text/.style={string type}]\data   %\pgfplotstablesave{\data}{pgfplotstempout.dat} \begin{tikzpicture} \begin{axis}[my stackbar plot]  \addplot table [x expr = \thisrow{--o},y=text] {\data};  \addplot table [x expr = \thisrow{-o}, y=text] {\data};  \addplot table [x expr = \thisrow{+o}, y=text] {\data};  \addplot table [x expr = \thisrow{++o}, y=text] {\data}; \end{axis} \end{tikzpicture} \quad \begin{tikzpicture} \begin{axis}[my stackbar plot]  \addplot table [x expr = \thisrow{--},y=text] {\data};  \addplot table [x expr = \thisrow{-}, y=text] {\data};  \addplot table [x expr = \thisrow{+}, y=text] {\data};  \addplot table [x expr = \thisrow{++}, y=text] {\data}; \end{axis} \end{tikzpicture}  \end{document} Schengen visa refused from German embassy What are all of the connected subsets of $\mathbb{Q}$? The answer are the singletons of $\mathbb{Q}$. I can show that the open intervals of $\mathbb{Q}$ are disconnected by choosing some irrational in the open set and using it to form a separation. But strangely enough, I am having a hard time seeing why the subset $\{p,q \}$ of $\mathbb{Q}$ is disconnected. If say $p &lt; q$ and we choose some irrational $p &lt; x &lt;q$, the separation of $\{p,q \}$ would, I think, be given by $ \{p\}$ and $\{q\}$. But these aren't open in $\mathbb{Q}$ ...? Should you declare methods using overloads or optional parameters in C# 4.0? I was watching , and it got me thinking about when optional parameters are available in C# what is going to be the recommended way to declare methods that do not need all parameters specified?  For example something like the FileStream class has about fifteen different constructors which can be divided into logical 'families' e.g. the ones below from a string, the ones from an IntPtr and the ones from a SafeFileHandle.  FileStream(string,FileMode); FileStream(string,FileMode,FileAccess); FileStream(string,FileMode,FileAccess,FileShare); FileStream(string,FileMode,FileAccess,FileShare,int); FileStream(string,FileMode,FileAccess,FileShare,int,bool);   It seems to me that this type of pattern could be simplified by having three constructors instead, and using optional parameters for the ones that can be defaulted, which would make the different families of constructors more distinct [note: I know this change will not be made in the BCL, I'm talking hypothetically for this type of situation].  What do you think? From C# 4.0 will it make more sense to make closely related groups of constructors and methods a single method with optional parameters, or is there a good reason to stick with the traditional many-overload mechanism? How can I install software or packages without Internet (offline)? How can I give a user one password for normal usage when logging in, and another password for system administration and sudo access? I want one user to have two passwords. How to capture the scene exactly as my eyes can see? Should a math paper use non-alphabetical ordering of authors' names for unequal contributions? Using alphabetical ordering for authors' names is a long established tradition in mathematics. In the situations of extremely unequal contributions, is it reasonable to break from this tradition?  Has there been any examples?  If it is common, what is the threshold for breaking from alphabetical ordering? What are the strategies for getting feedback on articles? As an academic, one has to publish. Often, an article is drafted, and you need feedback of a colleague.  I see that quite often some perspective articles are often written by long-term collaborating pairs of authors.  What are some strategies people use to get feedback on their articles  (outside of immediate boss; and in the case that none of the friends in the field of work in the specific domain of the article, and one needs specific (not general) feedback)?   How do you approach a colleague to simply read your article. Or do you just mention it at a conference to the most suitable colleague - would you like to read an article and give me feedback? How do handle the co-authorship or acknowledgement? Do you establish the limits at the "approach time"?  What are some strategies to establish a "publishing" duos (buddies). Sometimes, a question is perfectly solved by suggestion or actual resolution given in "comments" under it. In such case, how to mark that answer (comment) as "Accepted Answer"? Due to this, the question remains unanswered even if solution is provided in comment. What is the distribution of primes modulo $n$? How to make Gnome's "Activities" bar always visible?