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Prove that $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$   My proof is below. I am interested in other proofs.    My proof  Let   $$f (\mathrm X) := \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B)$$  Hence,  $$\begin{array}{rl} f (\mathrm X + h \mathrm V) &amp;= \mbox{tr} (\mathrm A (\mathrm X + h \mathrm V)^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A (\mathrm X ( \mathrm I + h \mathrm X^{-1} \mathrm V))^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I + h \mathrm X^{-1} \mathrm V)^{-1} \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I - h \mathrm X^{-1} \mathrm V + O (h^2)) \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) - h \, \mbox{tr} ( \mathrm A \mathrm X^{-1} \mathrm V \mathrm X^{-1} \mathrm B) + O (h^2)\\ &amp;= f (\mathrm X) - h \, \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) + O (h^2)\end{array}$$  Thus, the directional derivative of $f$ in the direction of $\mathrm V$ at $\mathrm X$ is  $$D_{\mathrm V} f (\mathrm X) = - \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) = - \mbox{tr} ( (\mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top})^\top \mathrm V ) = - \langle \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}, \mathrm V \rangle$$  and, lastly,  $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$ When given $p = 5, q = 11, N = 55$ and $e = 17$, I'm trying to compute the RSA private key $d$.  I can calculate $\varphi(N) = 40$, but my lecturer then says to use the extended Euclidean algorithm to compute $d$. That's where I get stuck.    Here's my work so far:  First I use the Euclid algorithm to calculate:  40 = 2(17) + 6  17 = 2(6) + 5 6 = 1(5) + 1 = gcd   So I know the GCD is 1. Applying the 'extended' section of the algorithm:  6 = 40-2(17) 5 = 17-2(6) 1 = 6-1(5) 1 = 6-1(17-2(6)) 3(6) = 1 (17) 3(40 - 2(17)) - 1(17) 3(40) - 3(17)   I know the answer is $33$, but I have no idea how to get there using the extended Euclidean algorithm. I can't figure out why I'm getting $3(40) - 3(17)$ when I know the answer should contain $33$. What is the adjectival form for the word "integrity?" I'm looking for the adjectival form of &quot;integrity.&quot; Instead of &quot;Be a person of integrity,&quot; I'd like to say something like &quot;Be [one word I'm looking for]&quot; I did a  for this, but I also wanted to know what stackexchange folks would like to say. What is the most efficient way to write the old-school:  StringBuilder sb = new StringBuilder(); if (strings.Count &gt; 0) {     foreach (string s in strings)     {         sb.Append(s + ", ");     }     sb.Remove(sb.Length - 2, 2); } return sb.ToString();   ...in LINQ? Easy way to change apache www location Is the word "management" singular or plural? Possible Duplicate:          Which one of the following is correct?     Management gets its ideas from its employees.   Management gets their ideas from their employees. How to choose which FaceTime number to call I have a friend with two iPhones, two different numbers.  When I call her with FaceTime audio, only phone A rings.  How can I call phone B?  The phones have two separate Apple ID addresses, but they are both set to receive Facetime calls on their 10 digit phone numbers.  I have both numbers listed in my contacts as 'Home' and 'Work' for the same person. How to change material's color in blender game engine? I was setting up up android-studio referring to . While installing oracle-java7-installer, this error is displayed:  sudo apt-get install oracle-java7-installer [sudo] password for cyan:  Reading package lists... Done Building dependency tree        Reading state information... Done oracle-java7-installer is already the newest version (7u80+7u60arm-0~webupd8~1). The following package was automatically installed and is no longer required:   snap-confine Use 'sudo apt autoremove' to remove it. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. 1 not fully installed or removed. After this operation, 0 B of additional disk space will be used. Do you want to continue? [Y/n] y Setting up oracle-java7-installer (7u80+7u60arm-0~webupd8~1) ... Downloading Oracle Java 7... --2017-11-27 08:40:47--  http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz Resolving download.oracle.com (download.oracle.com)... 223.223.158.138, 223.223.158.136 Connecting to download.oracle.com (download.oracle.com)|223.223.158.138|:80... connected. HTTP request sent, awaiting response... 302 Moved Temporarily Location: https://edelivery.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz [following] --2017-11-27 08:40:47--  https://edelivery.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz Resolving edelivery.oracle.com (edelivery.oracle.com)... 23.58.58.93, 2600:140f:7:18e::2d3e, 2600:140f:7:18d::2d3e Connecting to edelivery.oracle.com (edelivery.oracle.com)|23.58.58.93|:443... connected. HTTP request sent, awaiting response... 302 Moved Temporarily Location: http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz?AuthParam=1511752368_824528560c20825821ecbcac8f4e9be9 [following] --2017-11-27 08:40:48--  http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz?AuthParam=1511752368_824528560c20825821ecbcac8f4e9be9 Connecting to download.oracle.com (download.oracle.com)|223.223.158.138|:80... connected. HTTP request sent, awaiting response... 404 Not Found 2017-11-27 08:40:50 ERROR 404: Not Found.  download failed Oracle JDK 7 is NOT installed. dpkg: error processing package oracle-java7-installer (--configure):  subprocess installed post-installation script returned error exit status 1 Errors were encountered while processing:  oracle-java7-installer E: Sub-process /usr/bin/dpkg returned an error code (1)   How do I fix this error? I did a fresh installation of Ubuntu 16.04. By considering:  $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^1}{n^{2}} = \frac 1 2$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^2}{n^{3}} = \frac 1 3$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^3}{n^{4}} = \frac 1 4$$  Determine if this is true: $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}} = \frac 1 {{m+1}}$$  If it is, prove it.  If it is not, evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{{m+1}}$. Why do people sometimes use the past perfect form of a verb when the past would suffice? (e.g. "you had mentioned") Error message on Terminal launch shell: keep trailing newlines ('\n') in command substitution I want to be able to capture the exact output of a command substitution, including the trailing new line characters.  I realise that they are stripped by default, so some manipulation may be required to keep them, and I want to keep the original exit code.  For example, given a command with a variable number of trailing newlines and exit code:  f(){ for i in $(seq "$((RANDOM % 3))"); do echo; done; return $((RANDOM % 256));} export -f f   I want to run something like:  exact_output f   And have the output be:  Output: $'\n\n' Exit: 5   I'm interested in both bash and POSIX sh. Why is it not possible to overload class templates? Fixed points under action of conjugate group elements Lecture notes:  If $g$ and $h$ are conjugate then $fix(g)=fix(h)$. To see this, say $g=k^{-1}hk$  If $g.s=s \iff k^{-1}hk.s=s \iff h.(k.s)=(k.s)$  Me: I understand the last line- I just fail to see how it implies $fix(g)=fix(h)$  EDIT: As Ben S says I think there was a typo in the notes and they forgot modulus signs so- New Aim: show $|fix(g)|=|fix(h)|$ I heard about nautilus-gksu and would like to install it. Unfortunately, when I try, I get this:  glemi@XPS:~$ sudo apt-get install nautilus-gksu Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package nautilus-gksu   How and where can I get this package? which vectors are perpendicular to each other? which vectors are perpendicular to each other?  $\vec a = (1, -2, 3)$, $\vec b = (5, 4, 1)$, $\vec c = (1, 0, -5)$  Do i just take the dot product of 2 of them. If the dot product they are at $90^\circ$? But how do i know if there perpendicular? Let's say we have a cloud of dust which is a lightyear across and someone shoots a beam of light from point A to B , why it is not possible for an observer far far away to see the light while it travels through the cloud at the speed of light? Python: Find a substring in a string and returning the index of the substring Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K[x]$ imply relatively prime in $L[x]$?