Let $x, y \in \mathrm{R^n}$. Show that det$(I - xy^T) = 1 - y^{T}x$.  I test several cases for this problem, and it holds true when $n = 2, 3$, but when it comes to proof, I tried two ways and both failed:   Write $x, y$ explicitly in the form of $a_{ij}, b_{ij}$, and tried to find the relationship between the left and right side. But the left side could not be reduced to a relatively clearer form and turns out to add more complexity. Try to transform $I - xy^T$ into something similar to the right side.    I think the difficulty of this problem lies in the transformation from matrix $(I - xy^T)$ into scalar $1 - y^{T}x$, and I do not have a good way to make such transformation.  This question could be answered with matrix determinant lemma (the proof could be found . Even though this is pretty easy to understand, it lacks the reasoning process, which I find very unsatisfying.
Let be $u,v\in\mathbb{R}^n$, then $\det(I+uv^\intercal)=1+v^\intercal u $ where $I$ denotes the identity matrix of order $n$. How to prove this?   what I did:   let be $A=\{n\in\mathbb{N}: \forall u,v \in \mathbb{R^n}, \det(I+uv^\intercal)\neq1+v^\intercal u  \} $, and suppose $A\neq \varnothing $, then for the well-orderer-principle there exists $n_0\in A$ such $n_0\leq n,\;\forall n\in A$. since $1\notin A,\; n_0\neq 1 \rightarrow n_0-1\in\mathbb{N}\setminus A  $ so: $$\forall u,v\in\mathbb{R^{n_0-1}}: \det(I+uv^\intercal)=1+v^\intercal u $$ Then, let be $u,v\in\mathbb{R}^{n_0}$, so $$\det(I+uv^\intercal)= \displaystyle\sum_{j=1}^{n_0}(-)^{1+j}a_{1j}\det(A_{1j})$$ where $a_{ij}=\begin{cases} u_iv_j+1 &amp; i=j \\ u_iv_j &amp; i\neq j \end{cases}$, and $A_{1j}$ is the  submatrix of $I+uv^\intercal$ that results deleting the i-file and j-column. Then I try to open the term for $j=1$ and try to use the relation for the new matrix of $n_0-1$ order with the determinant because it holds the same form but I have other terms which difficult me the work. Do you know other method?.  Ps. , in page 2 I've found something similar but I don't understand what it means PS. I'm taking a course of numerical analisys
Why doesn't Zorn's lemma apply to $[0,1)$? Do small title edits constitute trivial edits? I would like to know how I can get the value of a node with the following paths:   config/global/resources/default_setup/connection/host config/global/resources/default_setup/connection/username config/global/resources/default_setup/connection/password config/global/resources/default_setup/connection/dbname   from the following XML:  &lt;?xml version="1.0"?&gt; &lt;config&gt;     &lt;global&gt;         &lt;install&gt;             &lt;date&gt;&lt;![CDATA[Tue, 11 Dec 2012 12:31:25 +0000]]&gt;&lt;/date&gt;         &lt;/install&gt;         &lt;crypt&gt;             &lt;key&gt;&lt;![CDATA[70e75d7969b900b696785f2f81ecb430]]&gt;&lt;/key&gt;         &lt;/crypt&gt;         &lt;disable_local_modules&gt;false&lt;/disable_local_modules&gt;         &lt;resources&gt;             &lt;db&gt;                 &lt;table_prefix&gt;&lt;![CDATA[]]&gt;&lt;/table_prefix&gt;             &lt;/db&gt;             &lt;default_setup&gt;                 &lt;connection&gt;                     &lt;host&gt;&lt;![CDATA[localhost]]&gt;&lt;/host&gt;                     &lt;username&gt;&lt;![CDATA[root]]&gt;&lt;/username&gt;                     &lt;password&gt;&lt;![CDATA[pass123]]&gt;&lt;/password&gt;                     &lt;dbname&gt;&lt;![CDATA[testdb]]&gt;&lt;/dbname&gt;                     &lt;initStatements&gt;&lt;![CDATA[SET NAMES utf8]]&gt;&lt;/initStatements&gt;                     &lt;model&gt;&lt;![CDATA[mysql4]]&gt;&lt;/model&gt;                     &lt;type&gt;&lt;![CDATA[pdo_mysql]]&gt;&lt;/type&gt;                     &lt;pdoType&gt;&lt;![CDATA[]]&gt;&lt;/pdoType&gt;                     &lt;active&gt;1&lt;/active&gt;                 &lt;/connection&gt;             &lt;/default_setup&gt;         &lt;/resources&gt;         &lt;session_save&gt;&lt;![CDATA[files]]&gt;&lt;/session_save&gt;     &lt;/global&gt;     &lt;admin&gt;         &lt;routers&gt;             &lt;adminhtml&gt;                 &lt;args&gt;                     &lt;frontName&gt;&lt;![CDATA[admin]]&gt;&lt;/frontName&gt;                 &lt;/args&gt;             &lt;/adminhtml&gt;         &lt;/routers&gt;     &lt;/admin&gt; &lt;/config&gt;   Also I want to assign that value to the variable for further use. Let me know your idea. If you shine a light away from the center of a Schwarzchild black hole, will it stay still? $$ A $$  Light is travelling away from the absolute center of the Schwarzchild black hole (i.e. none of that spinning/stretching field) at $\ C $, and so shouldn't $\ \Delta C$ need to be greater than $\ C $ in order for the light to collapse into the black hole, therefore violating the relativistic limit of $\ C $?    It seems like light would just 'hover' right in the center of a black hole then. Because if it goes away from it, the event horizon pulls it back, and if it goes into it, the velocity change must be greater than $\ C $ which violates general relativity; so technically there's no way for the light to move away, and so it would just be still.  The thing is that the light must move at the speed of light or general relativity doesn't work. It's massless... But in the center of a black hole, if the above is true (it can't move or escape), then it is not travelling at the speed of light. So it cannot be still.  It cannot be spinning in a constant orbit either, because that would require velocity in an outward angle. And it cannot move in an outwardly fashion, only in an inwardly one.  Is it spinning in an infinitely decaying spiral towards the center?  If so what happens to it if and when it reaches the direct and complete center, i.e. The case of it being 'shot' from the center.    $$ B $$  If you shot light at an angle towards a black hole, but not directly to the center of it  then that light from the outside will eventually become the same as the light that I said was shot from the center to the outside; because in both cases, they end up in an infinitely decaying spiral.    If so it would end up exactly like the light in the above paragraph, so then we need to figure out what happens to that light to figure this one out, or vice versa.    $$ C $$  If you shot light directly towards the center of a black hole it would have no spiral to decay into, so therefore it must stop at some point in the center of a black hole. If it stops though, it loses it's momentum and violates general relativity again because it's energy "disappears".    If so then it violates conservation laws. So I'm assuming this simply is not true.    Is the light travelling into a black hole somehow still travelling at the speed of light, in every direction possible where that direction keeps it facing towards the center?  If so, why is there not a point at which the only thing left for the light to do is stay still, since all other paths would increase the distance between it and the black hole's center, and therefore become Case A. And if there isn't, what would happen if there was?  Also, if it's a time and observation answer where it supposedly takes and infinite amount of time for C to hit the center, wouldn't it be the case that the black hole is 'eating' all of the available light in the universe and keeping it in stasis?  Also, what happens in a Kerr black hole then? Does it just rip the field of time/gravity at an angle such that the light would never get closer than the center? Is it feasible to transform each variable differently while doing multiple regression This question attempts to collect the few pearls among the dozens of bad C++ books that are published every year. Unlike many other programming languages, which are often picked up on the go from tutorials found on the Internet, few are able to quickly pick up C++ without studying a well-written C++ book. It is way too big and complex for doing this. In fact, it is so big and complex, that there are very many very bad C++ books out there. And we are not talking about bad style, but things like sporting glaringly obvious factual errors and promoting abysmally bad programming styles. Please edit the accepted answer to provide quality books and an approximate skill level — preferably after discussing your addition in . (The regulars might mercilessly undo your work if they disagree with a recommendation.) Add a short blurb/description about each book that you have personally read/benefited from. Feel free to debate quality, headings, etc. Books that meet the criteria will be added to the list.  Books that have reviews by the Association of C and C++ Users (ACCU) have links to the review. *Note: FAQs and other resources can be found in the  and under . Recently as per this       For individuals who have chosen to receive automatic updates through   Windows Update, we help upgradable devices get ready for Windows 10 by   downloading the files they’ll need if they decide to upgrade.   but Microsoft seems to be downloading hidden update in windows 7 :  according to this  :     An INQUIRER reader pointed out to us that, despite not having   'reserved' a copy of Windows 10, he had found that the ~BT folder,   which has been the home of images of the new operating system since   before rollout began, had appeared on his system. He had no plans to   upgrade and had not put in a reservation request.   As per the above two articles there was hidden update for windows 10 with respective to windows 7 and 8 machines ,whats the effective way to stop this hidden update?  right now i cant disable windows update since i wish to patch my system with latest updates,  i checked the question which was marked as possible  :     inorder to clarify how it differs it talks about the cancelled update   ,but this was latest issue which was araised two days before,i didnt   dig much about that and believe both were different ,apart from that   it was about after the cancellation of update a   nd this one was about preventing such an update from downloading to my pc  any suggestions to avoid this update? What does "had had" mean? How does this differ from "had"? For example, what is the difference between the following two sentences:     I had a bad day      I had had a bad day Can you nest html forms? Why does no one notice the Joker's presence as a policeman during the mayor's speech? I have a WinXP installation that is used to only connect to another PC with RDP.   A dummy user account gets logged in automatically, then the user must click a link to start the RDP connection with predefined parameters.  Is there a way to run the RDP client without Windows Explorer being accessible? Calculating the optimal number of bins for severely skewed data A conceptual doubt regarding Forced Oscillations and Resonance While studying about the Resonance and Forced Oscillations, I came across a graph in my textbook that is given below:-  Now, the author writes      As the amount of damping increases, the peak shifts to lower frequencies.   Why does this happen? And does this imply that at higher damping levels one cannot achieve a higher amplitude by setting the period of the forced oscillation to be equal to the natural frequency? This seems strange because, the highest amplitude is achieved when the natural frequency is equal to the driving frequency. But, I guess that the rules are somehow different for the damping case. Not able to execute a .sh file: /bin/bash^M: bad interpreter I wanted to execute a shell script:  -rwxr-x--x 1 root root   17234 Jun  6 18:31 create_mgw_3shelf_6xIPNI1P.sh   I tried to do a standard procedure, but I got this error:  ./create_mgw_3shelf_6xIPNI1P.sh  localhost 389 -l /opt/fews/sessions/AMGWM19/log/2013-06-06-143637_CLA-0  DEBUG   cd/etc/opt/ldapfiles/ldif_in ; ./create_mgw_3shelf_6xIPNI1P.sh  localhost 389 -l /opt/fews/sessions/AMGWM19/log/2013-06-06-143637_CLA-0 **ERROR  sh: ./create_mgw_3shelf_6xIPNI1P.sh: /bin/bash^M: bad interpreter: No such file or directory**   What does it mean? I was doing this as the root user under the root group.  Does it mean that the file does not have the correct permission for the root user? The following is exercise 8.2 in 3rd edition (and exercise 8.19 in 2nd edition) of Goldstein's Classical Mechanics.   Adding the total time derivative of a function of $q_i$ and t to the Lagrangian will not change the the Euler-Lagrangian equation. So if we make the following change to Lagrangian,   $$L'(q,\dot{q},t) = L(q,\dot{q},t) + \frac{dF(q_1,q_2,...,q_n,t)}{dt}$$ we can get  $$ \frac{d}{dt}\frac{\partial{L'}}{\partial{\dot{q_i}}} - \frac{\partial{L'}}{\partial{q_i}} = 0 $$ from  $$ \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q_i}}} - \frac{\partial{L}}{\partial{q_i}} = 0 $$  How can we get the corresponding Hamiltonian equation part? This is to prove  $$ \dot{p'_i} = \frac{\partial{H'}}{\partial q_i} $$ $$ -\dot{q_i} = \frac{\partial{H'}}{\partial p'_i}  $$  from   $$ \dot{p_i} = \frac{\partial{H}}{\partial q_i} $$ $$ -\dot{q_i} = \frac{\partial{H}}{\partial p_i}  $$ where $p'_i = \frac{\partial L'}{\partial \dot q_i}$.  Edit  The corresponding $H'$ is $$ H' = \sum_k{p'_k \dot{q_k}} - L' $$ where $p'_k = \frac{\partial L'}{\partial \dot q_k}$. How can one show that for triangles of sides $a,b,c$ that   $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} &lt; 2$$  My proof is long winded, which is why I am posting the problem here.   Step 1: let $a=x+y$, $b=y+z$, $c=x+z$, and let $x+y+z=1$ to get   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}&lt;2$  Step 2: consider the function $f(x)=\frac{1-x}{1+x}$, and note that it is convex on the interval (0,1), so the minimum of   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}$  is reached when the function takes the extreme points. i.e. $x=y=0, z=1$.    But going back to the fact that this is a triangle, we note that $x=y=0 \implies a=0$ which is not possible, so the inequality is strict. I have searched the web a lot and could not find an answer that fits my situation. Here are two links that are quite contradictory:       My situation:  I've visited Portugal in December 2012 for 29 days on a Schengen visa valid from December 1st, 2012 to February 1st, 2013 for research purposes (I am a Ph.D. student). I want to visit Sweden from February 10th, 2013 to May 10th, 2013 (89 days). I have applied for a Schengen visa again (for research). How will the 90/180 rule work in my situation? Techniques for embedding images with transparent backgrounds in LaTeX Let $A$ and $B$ be two real $n\times n$ matrices such that $AB-BA=A$ Prove that $A$ is not invertible and that $A$ is nilpotent.  My attempt is the following. It holds that $AB=(B+I)A$ If $A$ were invertible, $B=A^{-1}(B+I)A$. Taking trace on both sides yields $tr(B)=tr(B)+n$, hence $n=0$, which is a contradiction. I can't prove that $A$ is nilpotent though. Are token creatures considerd as a creature spell?