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A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator.   In this question is it necessary to define what the projection operator is? And won't the eigenvalue just be zero? 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,  $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$ How to change the href for a hyperlink using jQuery How can you change the href for a hyperlink using jQuery? I have always heard the term used in referring to a single word. When browsing questions on this site, I've seen it used applied to entire phrases, and have suppressed the compulsion to edit them and replace the term with origin. Is it possible to identify this chrysalis? If $f^3=\rm id$ then it is identity function Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f^3(x)=f\circ f\circ f(x)=x$ for all $x$.  How can I prove $f$ is the identity function? How to change the starting reference number in Natbib in a given bibunit? I'm using bibunit and Natbib packages with a numerical numbering.   \begin{bibunit}[plainnat]     \nocite{ref1,ref2}     \putbib[mybiblio] \end{bibunit}  \begin{bibunit}[plainnat]     \nocite{ref3,ref4}     \putbib[mybiblio]  \end{bibunit}   How can I set the starting number for ref3 and ref4 to be different from 1 (say 15), without affecting the starting number for ref1 and ref 2? The desired result is the following:   [1] ref 1 [2] ref 2 [15] ref3 [16] ref4 Does anyone know any example that invalidates the following affirmation:     If a morphism $f:A\to A$ induces the identity $\hat f:\operatorname{Spec} \left( A \right) \to \operatorname{Spec} \left( A \right)$ then $f = \operatorname{id} _A $. What exactly is the point of memoryview in Python How can I remove malicious spyware, malware, adware, viruses, trojans or rootkits from my PC? Prove no existing a smooth function satisfying ... related to Morse Theory how is this possible? I know there is some trick, should someone please explain?! How to disable battery charging? From the pmset man page:     POWER SOURCE ARGUMENTS      -g with a 'batt' or 'ps' argument will show the state of all attached power sources.      […] Certain Apple battery calibration tools may temporarily disable   battery charging, even though your Apple portable is plugged into AC   power.      Such tools will never surprise you with this behavior; you will know   it's happening. This state will be reflected in an additional line of   output to 'batt' or 'ps' readings, including the pid of the app   inducing this behavior.       'ChargeInhibit':    1432    So, does any piece of publicly available software exist to force the battery not to charge? Importing coordinate data from Excel into QGIS I have an Excel file containing data collected over many years as latitude and longitude coordinates for archaeological sites. How can I import it into QGIS as a new layer? Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer. Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}$ is an integer.   I'm not familiar to factorial and I don't have much idea, can someone show me how to prove this? Thank you. Duplicate application icons in Ubuntu dock upon launch I'm using Ubuntu 17.10 with GNOME shell and Numix-Circle icon theme.   When I hover over the Numix icon for Nightly Web Browser, which is pinned to the dock I get the title "Nightly Web Browser".      But when I launch the program, another icon pops up at the bottom, with the name "Nightly". Why is a black hole black? In  (ignoring Hawking radiation), why is a  black? Why nothing, not even light, can escape from inside a black hole? To make the question simpler, say, why is a  black hole black? Countable or uncountable; family of disjoint triangles? If we say that $\mathcal{S}$ is a family of disjoint triangles on a plane, is $\mathcal{S}$ a countable or an uncountable set?  I believe that it is an uncountable set since $\mathbb{R}$ is uncountable and our triangles' corner coordinates are $\in \mathbb{R}$, but I cannot prove this formally, any tip would help. I am moving from Malaysia to the US, and planning to bring my desktop computer with me. I plan to pack it in my checked-in luggage. Will the computer risk being damaged? Any precaution should I take to reduce this risk? Will there be any trouble with the airline or immigration about bringing computer? I have been using LaTeX for a long time now, and I find myself often going to this website to solve my problems. This often includes copy pasting codes which include \newcommand, \newenvironment and \def. I think it is time I start to learn how to use them so I can solve my problems myself! It is always better if you actually know what is going on inside your code.  I am wondering if there is any turtorial or book out there that handles how to use those commands. Or is the best way to learn to just mess with a random file?   Thanks in advance!  (Also this is my first time I post here after using this website for so long, I really appreciate everyone who posts and tries to help other people here!)