Prove rank(λI−A)=rank((λI−A)^2). Given that A is a diagonalizable matrix and λ an eigenvalue of A.
Let $A$ be a diagonalizable matrix and $\lambda$ be an eigenvalue of A. Prove that rank($\lambda I - A$) = rank(($\lambda I - A)^2$).
I am trying to prove this inequality  $$|e^z-1|\leq e^{|z|}-1\leq |z|e^{|z|}$$  I've tried calculating the difference in their power series   $$\sum_{k=0}^\infty\frac{|z|^k}{k!}-1-\left|\,\sum_{k=0}^\infty\frac{z^k}{k!}-1\,\right|$$   but did not get anywhere. Covariance of Euler-Lagrange equations under change of generalized coordinates My understanding is that heat is essentially atomic vibrations. If there are almost no atoms, how can there be residual heat?  Also, as I understand in space there is no heat transfer via convection and it is mainly through EM radiation. In this case, why is it that if i were to go into space with no suit and just an airtight helmet through which I could breathe, I would still freeze to death very quickly? Does "Going Infinite" necessarily end the game? Windows 7 Audio Too Loud In Windows 7 (64-bit, Ultimate) the default audio output is extremely high. This has been an issue for a while but I was able to get around it as my previous headphones took more power to drive so it actually worked in my benefit.  However, now I have a set of really sensitive headphones and they're being over-powered. So much so that if I put it down at around 5% of total output it's bearable.  Is there a way I can sort of limit the system's volume so that it's incapable of going extremely loud and then have the rest of it just behave as normal?  Things I've tried:   Uninstalling and re-installing the audio device and drivers. Limiting each application individually. Searching for third-party software (so far nothing looks promising).   The first has no effect. The second works find but I have to do it for every application individually (which can be a pain) and many of them reset their volume every time they're launched, which makes it almost useless. Some even reset the volume while I'm playing a game!  So does anyone have any suggestions on how to make my volume reasonable?  The device is, apparently, a Cirrus Logic CS4206A (AB 24). The computer is an 2010 iMac. UK PLAB Visa Refusal due to unexplained bank deposits Seeking world map with boundaries of regions and communities for commercial use? This question is NOT about access to USB or external drives.  WPS Witer and other just installed Ubuntu software can NOT open files saved on my /media directory. I keep getting the annoying message: « Could not read the contents of '/media': Permission denied ».   MY SIMPLE QUESTION REMAINS:  How can I access all files and folders on my user username/media/ directory?  I'm using Ubuntu 18.04.3 LTS.  Ubuntu's Nautilus can indeed access all files and folders in all locations. However, Ubuntu's software that I have downloaded (e.g. KRITA, WPS WRITER etc) can NOT access any file or folder on the /media directory. Thanks in advance. Removing trailing newline character from fgets() input Windows 7 finding location of installed program How does gravity escape a black hole? How do I change font size on the DataGridView? How do I add polygons to fill a face? trying to recreate this Elijah bottle. I have two sides, one of which is outlined in red. How do I fill that face with polygons so I can then grab the center (that red square dot) to pull it outward thereby creating the slight outward bowing of the bottle? How to use different domains for multilingual sites and keep users logged in? I have a site that is in French and English. I am able to successfully setup so that users can switch back and forth between the two domains to view either language, however I cannot seem to keep users logged in. When they are logged into the english domain and then switch to the french domain they are required to login again. However if they switch back to the english domain they are still logged in.  As an example we'll use www.english-domain.com as the base (default) domain and www.french-domain.com as the secondary domain  In settings.php I have modified the following, however it doesn't seem to be keeping them logged in. Am I missing something?  /**  * Drupal automatically generates a unique session cookie name for each site  * based on its full domain name. If you have multiple domains pointing at the  * same Drupal site, you can either redirect them all to a single domain (see  * comment in .htaccess), or uncomment the line below and specify their shared  * base domain. Doing so assures that users remain logged in as they cross  * between your various domains.  */ $cookie_domain = 'english-domain.com';   Note: That I have also flushed both site and browser cache as well as deleted any cookies left on my machine.  Update:  Here are my language settings: In my class design, I use abstract classes and virtual functions extensively. I had a  feeling that virtual functions affects the performance. Is this true? But I think this performance difference is not noticeable and looks like I am doing premature optimization. Right? How does find -name work? C++ STL Vector Iterator accessing members of an Object I think I've declared a Vector with an object correctly.  But, I don't know how to access it's members when looping with Iterator.  In my code, the line --->>  cout &lt;&lt; " " &lt;&lt; *Iter;  How do I print the contents of the members?  Like *Iter.m_PackLine ???  Not sure if I used the correct terminology, but appreciate the help!  Thanks  class CFileInfo {   public:       std::string m_PackLine;       std::string m_FileDateTime;       int m_NumDownloads; };  void main() {   CFileInfo packInfo;    vector&lt;CFileInfo, CFileInfo&amp;&gt; unsortedFiles;   vector&lt;CFileInfo, CFileInfo&amp;&gt;::iterator Iter;    packInfo.m_PackLine = "Sample Line 1";   packInfo.m_FileDateTime = "06/22/2008 04:34";   packInfo.m_NumDownloads = 0;   unsortedFiles.push_back(packInfo);    packInfo.m_PackLine = "Sample Line 2";   packInfo.m_FileDateTime = "12/05/2007 14:54";   packInfo.m_NumDownloads = 1;   unsortedFiles.push_back(packInfo);   for (Iter = unsortedFiles.begin(); Iter != unsortedFiles.end(); Iter++ )  {     cout &lt;&lt; " " &lt;&lt; *Iter; // !!! THIS IS WHERE I GET STUMPED     // How do I output values of the object members?   } }  // end main When abstract logic circuits get translated to the electrical equivilant, how are conversions from 0 to 1 handled? I've searched around and there's lots of good information out there, but I'm a bit of a n00b when it comes to the precise meanings of words like current, voltage, amplify, resistor, diode etc. So what at first presents itself as a simple explanation quickly turns into a wikipedia tabfest of ever increasing complexity. With that in mind please forgive my probably inaccurate use of these terms in this post.  The basic problem for me is that gates on a logic circuit appear to be creating power from nothing, which is impossible. For example imagine two NOT gates connected in sequence: To my uninitiated mind what seems to be happening is that the first NOT gate cuts the power to the second NOT gate, which responds by magically conjuring new power out of thin air and sending it on it's way. The only explanation I can come up with feels a bit redundant and absurd (That every single gate on the diagram is implied to be directly connected to a power source)  The closest I've come to an answer is something to do with changing the "signal" from "high" to "low" but again, I'm not familiar enough with the language. It still suffers from the same problem, if you've got a low signal how is it that a NOT gate can turn it into a high one without directly drawing on a power source?  To wrap up and present the question in all it's rookie glory: How do logic gates physically convert 0 to 1 without violating the law of conservation of energy? BAV99 forward voltage I am reading the  From the table on ELECTRICAL CHARACTERISTICS (page 2) (TA = 25°C unless otherwise noted) (Each Diode)     Forward Voltage mVdc     IF = 1 mAdc 715mV   IF = 10 mAdc 855mV   IF = 50 mAdc 1.0V   IF = 150 mAdc 1.25V     On page 3 there is the following VI curve graph:  Clearly at 25°C - 100mA is Vf = 0.9V. There is huge difference between the table and the IV curve.  Which source of information I should trust? Is this (typical) and (maximum) where the table guarantees the diode won't exceed Vf 1.0v at 50mA while the graph is just typical values? $H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$ Given the following definition of normal subgroup     A subgroup $H$ of a group $G$ is said to be normal if, for every $g\in G$: $$gH=Hg$$   I've tried to show that $H\mathrel{\unlhd} G$ if and only if we have $gHg^{-1}=H$ for every $g\in G$:  (if) Let $H\mathrel{\unlhd} G$, $g\in G$; if $l\in gHg^{-1}$, there exists $h\in H$ such that $l=ghg^{-1}$ and since $gh$ = $h'g$ for some $h'\in H$, then $l=ghg^{-1}=h'(gg^{-1}) = h'\in H$. Let $h\in H$; we can write $h$ as $hgg^{-1}$, and since $hg = gh'$ for some $h'\in H$, we obtain $h = hgg^{-1} = gh'g^{-1}\in ghg^{-1}$. So $gHg^{-1}=H$.  (only if) Let $gHg^{-1}=H$, $g\in G$; if we take an $l\in Hg$, $l = hg$ for some $h\in H$, we note immediately that $l=hg=(gh'g^{-1})g=gh'\in gH$. What should I do to show that $gH\subset Hg$?  I'm also wondering if this mess is correct, and if it is possible a more elegant proof of this fact.