I wonder the condition of A to make the following statement true: $\det(e^{tA})=e^{t\cdot{tr} (A)}$, where A is an $n\times n$ matrix. If A is diagonalizable, then the statement can be proved. So, will the statement holds for A which isn't diagonalizable? I have tried using the Jordan form, but it is too complicated to obtain the result. Thank you!
Suppose $A$ is a $n \times n$ constant matrix. How can I prove $\det(e^A) = e^{\displaystyle \sum_{\lambda_i\in\sigma(A)} \lambda_i}$,  where $\sigma(A)$ is the multiset of eigenvalues of $A$?  The following matlab code shows this is true in $n = 3$ case:  A = rand(3) detA = exp(sum(eig(A))) detmA = det(expm(A))
Ubuntu does not offers me "install alongside Windows" Any of the partitions makes in Windows was not recognised by Ubuntu, or Zorin OS, or LinuxMint. It's like a blank hard disc, 250 gb free. I'm a new Linux user, but what confuses me is that I was able to install Ubuntu alongside Windows on my laptop with no problems, but on PC... no way! Suppose $G$ has odd order, show the function $f:G\rightarrow G$ defined by $f(x)=x^2$ is injective. This proposition is easily provable if we assume $G$ is Abelian, but I don't know how to start this without the assumption of being Abelian. Why, historically, do we multiply matrices as we do? Multiplication of matrices &mdash; taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product &mdash; is not a very intuitive operation: if you were to ask someone how to mutliply two matrices, he probably would not think of that method. Of course, it turns out to be very useful: matrix multiplication is precisely the operation that represents composition of transformations. But it's not intuitive. So my question is where it came from. Who thought of multiplying matrices in that way, and why? (Was it perhaps multiplication of a matrix and a vector first? If so, who thought of multiplying them in that way, and why?) My question is intact no matter whether matrix multiplication was done this way only after it was used as representation of composition of transformations, or whether, on the contrary, matrix multiplication came first. (Again, I'm not asking about the utility of multiplying matrices as we do: this is clear to me. I'm asking a question about history.) I lost a couple hundred lines of a Vim plugin I wrote and continue back up every so often because I extracted a tar file containing an older version. I decided not to use source control management when I wrote the plugin, since I only update it periodically.  Why doesn't tar ask before overwriting existing files by default? I'm guessing that there's some rationale behind why the default behavior is to overwrite files silently, since we usually want safety over speed and convenience. For example, we usually use #ifdef NDEBUG, not #ifdef DEBUG; similarly, I would expect tar ask me before overwriting any files, and for it to have some flag which I can use to tell it to overwrite files without asking me. Also, how can make it so that tar does ask me by default, without me having to give it any flags? I'm fine with using tools other than tar, if they support this safer default behavior.  Thank you for your time! Many apps are missing from Ubuntu Software app I tried to enable additional repositories according to the instructions from , however my Ubuntu Software app is still missing a lot of popular apps. Is there anything I'm missing?  Ubuntu 16.10 Ubuntu Software app 3.20.1 How can I use asymmetric encryption, such as RSA, to encrypt an arbitrary length of plaintext? RSA is not designed to be used on long blocks of plaintext like a block cipher, but I need to use it to send a large (encrypted) message.  How can I do this? Escape a string for a sed replace pattern Is there a way to access Windows file with administrative rights using WSL?  sudo vi /mnt/c/Users/a/admin/file.txt Story about space explorers who communicate but then decide they must destroy each other I'm looking for a sci-fi story or novel from many years ago where a deep space explorer encounters an alien explorer. The explorers learn to communicate; but on parting, each decides to destroy the other in order to protect his own world. This is a  about Connection Refused    We see a lot of questions to the effect      When I try to connect to a system I get a message      Connection refused      Why is this ? Distorted circles in axis environment with TikZ With a fresh installation of TeX Live (updated just before writing this post), circles in axis environments are distorted. Consider the following MWE:  \documentclass{article} \usepackage{pgfplots} \pgfplotsset{compat=1.16}  \begin{document}  \begin{tikzpicture} \begin{axis}[   ymin=-3, ymax=3,   xmin=-3, xmax=3 ]  \draw (1,0) circle [radius=1.5]; \fill (1,0) circle [radius=1];  \end{axis} \end{tikzpicture}  \end{document}   The output looks like this:   Obviously, the circles aren't even circles. I haven't encountered this problem before. Did I do anything wrong while setting up my new system?  EDIT: It's not an issue of using axis equal or not since the axes are scaled more or less equally already. Instead, the circles appear as rotated ellipses that are strongly stretched out in one dimension.  EDIT 2: Here's the output of \listfiles:  article.cls    2019/12/20 v1.4l Standard LaTeX document class   size10.clo    2019/12/20 v1.4l Standard LaTeX file (size option) pgfplots.sty    2018/03/28 v1.16 Data Visualization (1.16) graphicx.sty    2019/11/30 v1.2a Enhanced LaTeX Graphics (DPC,SPQR)   keyval.sty    2014/10/28 v1.15 key=value parser (DPC) graphics.sty    2019/11/30 v1.4a Standard LaTeX Graphics (DPC,SPQR)     trig.sty    2016/01/03 v1.10 sin cos tan (DPC) graphics.cfg    2016/06/04 v1.11 sample graphics configuration   pdftex.def    2018/01/08 v1.0l Graphics/color driver for pdftex     tikz.sty    2020/01/08 v3.1.5b (3.1.5b)      pgf.sty    2020/01/08 v3.1.5b (3.1.5b)   pgfrcs.sty    2020/01/08 v3.1.5b (3.1.5b) everyshi.sty    2001/05/15 v3.00 EveryShipout Package (MS) pgfrcs.code.tex  pgfcore.sty    2020/01/08 v3.1.5b (3.1.5b)   pgfsys.sty    2020/01/08 v3.1.5b (3.1.5b) pgfsys.code.tex pgfsyssoftpath.code.tex    2020/01/08 v3.1.5b (3.1.5b) pgfsysprotocol.code.tex    2020/01/08 v3.1.5b (3.1.5b)   xcolor.sty    2016/05/11 v2.12 LaTeX color extensions (UK)    color.cfg    2016/01/02 v1.6 sample color configuration pgfcore.code.tex pgfcomp-version-0-65.sty    2020/01/08 v3.1.5b (3.1.5b) pgfcomp-version-1-18.sty    2020/01/08 v3.1.5b (3.1.5b)   pgffor.sty    2020/01/08 v3.1.5b (3.1.5b)  pgfkeys.sty pgfkeys.code.tex  pgfmath.sty     pgfmath.code.tex pgffor.code.tex tikz.code.tex l3backend-pdfmode.def    2020-02-03 L3 backend support: PDF mode supp-pdf.mkii epstopdf-base.sty    2020-01-24 v2.11 Base part for package epstopdf epstopdf-sys.cfg    2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live Brownian motion - Hölder continuity Let $B$ stand for a Brownian motion on a finite interval $[0,1]$. If I am not wrong, I think that there exists a positive constant $c$, such that almost surely, for $h$ small enough , for all $0&lt; t &lt; 1- h$  \begin{align} |B(t+h)-B(t)| &lt; c\sqrt{h\log(1/h)} \end{align} or something like this. As a result   \begin{align} \bigg|\frac{B(t+h)-B(t)}{h}\bigg| &lt; K(h)  \end{align}  Am I correct ? What are the criteria for posts to be included in "Top network posts" on new profile page? From the name of it, certainly it looks for posts with the highest scores. However, recently I found that some of the posts are not listed. Example: My  (not sure if it's that good, just because hot network question) was included in some point, but then it's not listed on . I guess it's because the question was closed, e.g as a dupe? (I don't have any hidden communities on the settings)    However, it's listed normally on .    Also, what other criteria that makes the posts to be included/excluded in &quot;Top network posts&quot;? How to connect Nokia Lumia 730 to Ubuntu 15.04? My Nokia Lumia 730 (WP8.1) is not getting detected on Ubuntu 15.04.  Can anybody tell me how to fix this? How to add a description to the user name input field in the registration form? I want a description under the user name input field in the Wordpress registration form like in the next example. How can I do this?  //create an extra field in login page for invitation code add_action( 'register_form', 'add_invitation_code_field' );  function add_invitation_code_field() {    ?&gt;       &lt;p&gt;          &lt;label for="invitation_code"&gt;             Invitation code             &lt;input id="invitation_code" class="input" type="text" size="20" value="" name="invitation_code"/&gt;             &lt;span class="description"&gt;&lt;?php _e("Description text"); ?&gt;&lt;/span&gt;          &lt;/label&gt;       &lt;/p&gt;    &lt;?php } Idempotents in a ring without unity (rng) and no zero divisors. Question: Given a ring without unity and with no zero-divisors, is it possible that there are idempotents other than zero?   Def: $a$ is idempotent if $a^2 = a$.   Originally the problem was to show that $1$ and $0$ are the only idempotents in a ring with unity and no zero-divisors, but I wonder what happens if we remove the unity condition.   I am trying to find a ring with idempotents not equal to $0$ or $1$. So far my biggest struggle has been coming up with examples of rings with the given properties.   Does anyone have any hints? How should I attack this problem? Is it correct to say user accounts or users account when referring to the accounts any user has on a site like this one?  In general, in the case of a noun that is used as adjective for the noun that follows, is it better to use &lt;plural-noun&gt; &lt;singular-noun&gt; or &lt;singular-noun&gt; &lt;plural-noun&gt;? I was wondering how light or any electromagnetic radiation can be "absorbed" if it consists of electric and magnetic fields. For example if there is a charge at point A, and the  light reaches point A, it will cause that charge to move depending on the value of the electric field. However, why wouldn't this electric field continue on and influence others particles everywhere else? For example, if you have a static charge creating an electric field, then if you put a charge at point A it will feel its effects and then if you bring a charge at another location B it will also feel its effects, namely it will not have been absorbed. Please shed some light on this. Thanks  edit: Does this mean the electron that absorbs this light moves in such a way that it creates an electric field of its own canceling the initial one? Or am I way off? What part of speech does here have in the following sentence?     I am here.   I say that in that sentence, here must be an adverb because:    It modifies the verb am by describing where I am. Am is a “being” verb in this instance, not a “linking” verb.   My friend contends that here is a noun because the word here is, in this instance, defined as this place, which in Merriam-Webster is the definition for the noun here. Durrett Example 1.9 - Pairwise independence does not imply mutual independence?