If we know that $\lambda_1$ and $\lambda_2$ are eigenvalues of matrix $A=\begin{bmatrix} a &amp; b \\ c &amp; d \end{bmatrix}$ prove that then $\lambda_1^2$, $\lambda_2^2$ and $\lambda_1\lambda_2$ are eigenvalues of $B=\begin{bmatrix} a^2 &amp; ab &amp; b^2 \\ 2ac &amp; ad+bc &amp;2bd \\ c^2 &amp; cd &amp;d^2\\ \end{bmatrix}$  I tried to calculate eigenvalues of matrix A and matrix B and then to compare them, but it is way to complicated to calculate, is there any easier solution to this?
This is the problem I am currently having trouble with:  If $\lambda_1$ and $\lambda_2$ are eigenvalues of matrix  $$         \begin{bmatrix}         a &amp; b\\         c &amp; d\\         \end{bmatrix}$$prove that $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$ are eigenvalues of matrix $A$.     $$A=\begin{bmatrix}     a^2 &amp; ab &amp; b^2\\     2ac &amp; ad+bc &amp;2bd\\     c^2 &amp; cd &amp;d^2\\     \end{bmatrix}$$  What have I considered so far:   I tried to find characteristic polynomial of first matrix and then find the roots which would be $\lambda_1$ and $\lambda_2$. I was hoping I can make a connection with characteristic polynomial of matrix $A$, but it got too complicated and did not seem like the right idea. Since there are two eigenvalues for first matrix, there are two eigenvectors. I thought about somehow using $A\vec v_i=\lambda_i\vec v_i$, where $i=1,2$  Also, I thought about finding a connection between these two matrices. What I see is that the sum of elements of first and second column, of first matrix, are $(a+c)$ and $(b+d)$, respectively.  Sum of first, second and third column of matrix $A$ are $(a+c)^2, (a+c)(b+d)$ and $(b+d)^2$. Now, if I could assign $\lambda_1$ and $\lambda_2$ to $(a+c)$ and $(b+d)$, then I would get that sum of first, second and third column of matrix $A$ are $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$, but what would be next?   I thought it would be good to combine 2. and 3rd idea, but I am not making any progress.  I appreciate any help.
Helpful flags not getting updated after getting 'close question' privilege Possible Duplicate:          I recently got the ability to cast close votes. So, even if I flag a question as 'not constructive' or 'not a real question', it is still considered as a close vote. What is strange is that my 'helpful flag' count has stopped increasing after this change. Or at least it doesn't increase in cases where my flagging was automatically changed to a close vote and the post actually was closed. Is this how it is supposed to be?   I am guessing that once the post gets closed, the moderators are probably not bothered about marking flags as helpful. The only problem is that the Deputy badge will be much more difficult to get now. Earlier all the flags were getting converted to helpful, but now they just go in as close votes and don't show up on my profile page. replace internal memory Is it possible to replace internal memory of a android phone ?  How difficult it is?  Is internal memory part of System On Chip itself, so that it is next to impossible to remove them ?  Have anybody tried this out ?  My device is "Redmi Note" - but you can consider this as a generic question. If these is any device where replacing internal memory is  possible, I will switch to that device :)  Thanks I am trying to prove the following identities:  a. $$\sum_{k=0}^n(-1)^k{n\choose k}^2 = \bigg\{^{0 \ \text{if k is odd}}_{(-1)^m{2m\choose m} \ \text{if n = 2m}}$$ b. $$\sum^k_{i=0} {n+i \choose i}={n+k+1\choose k}$$  For part $b$ I know I can use induction. It is clearly true when $k=0$ and then assuming for $k-1$ we get $$\sum^k_{i=0}{n+i\choose i}={n+k\choose k-1}+{n+k \choose k}={n+k+1\choose k} \ ...$$ gpgkeys: key C927EBE00F1B0520 not found on keyserver Possible Duplicate:    I am using 10.04 LTS - Lucid Lynx. I shutdown the computer while a software update was running. Then after I restarted my computer and while the update manager was running, I got this error:  Could not download all repository indexes The repository may no longer be available or could not be contacted because of network problems. If available an older version of the failed index will be used. Otherwise the repository will be ignored. Check your network connection and ensure the repository address in the preferences is correct. GPG error: http://toolbelt.herokuapp.com ./ Release: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY C927EBE00F1B0520 Failed to fetch  http://ppa.launchpad.net/rabbitvcs/ppa/ubuntu/dists/**DISTRIBUTION**/main/binary-i386/Packages.gz 404  Not Found   I went through the answers to  and I tried this $ sudo launchpad-getkeys  Please wait... launchpad-getkeys is running an update so it can detect the missing GPG keys Trying to import all the missing keys gpg: requesting key 0F1B0520 from hkp server keyserver.ubuntu.com gpgkeys: key C927EBE00F1B0520 not found on keyserver gpg: no valid OpenPGP data found. gpg: Total number processed: 0 launchpad-getkeys has finished importing all missing GPG keys.  Try running sudo apt-get update - you shouldn't see any key  errors anymore  But when I run sudo apt-get update again, it again shows that the key was not found on the keyserver. How to solve this problem? Missing links to responses to comments on nomination posts In the responses tab in my profile, responses to comments on nomination posts include a working link in the page source, but because there's no title the links are empty and useless.    When listed in the inbox such posts do have a title, though in this case the link is to the post and not directly to the comment. (Actually, the links in responses end up being redirected and also go to the post and not to the comment, but they try to go to the comment.)  In Supercollider:    In Global Profile: Dynamic grid with Atlas generation To give some background, I am attempting to solve the problem from Artin's algebra textbook which was already asked in this link:  Now, I would like to prove the following two claims which are not obvious to me. If I can prove these, I will be able to show that $\zeta_5\notin \mathbb{Q(\zeta_7)}$ as needed. I suspect that both claims will have similar proofs.  Claim 1: $\zeta_5$ has degree $4$ over $\mathbb{Q}$ Claim 2: $\zeta_7$ has degree $6$ over $\mathbb{Q}$  Attempt to prove Claim 1 Let $α=\zeta_5$. I would like to find the monic irreducible polynomial for $α$, but unfortunately I do not know what it is yet. I see $α=cos(\frac {2π}5)+isin(\frac {2π}5)$ and $α^4=cos(\frac {4π}5)+isin(\frac {4π}5)$. Please help me to find the irreducible polynomial, and (ideally) to show that it is irreducible. Is there a way for non-root processes to bind to "privileged" ports on Linux? It's very annoying to have this limitation on my development box, when there won't ever be any users other than me.  I'm aware of , but none of them do exactly what I want:    (The version in Debian testing, 1.0, only supports IPv4)  (the "nat" table is not yet implemented for ip6tables, the IPv6 version of iptables) sudo (Running as root is what I'm trying to avoid) SELinux (or similar). (This is just my dev box, I don't want to introduce a lot of extra complexity.)   Is there some simple sysctl variable to allow non-root processes to bind to "privileged" ports (ports less than 1024) on Linux, or am I just out of luck?  EDIT: In some cases, you can  to do this. The tablet functionality stops working after I wake my Lenovo Tablet from sleep mode (closing &amp; opening lit). The error message is "Tablet Serial Wacom Tablet (Lenovo X201) removed" (screenshot: ). In the system settings under input devices it states that "No tablet device is detected". Only a restart helps and the touch screen works again.  This error appeared only after upgrading from Kubuntu 12.04 to 14.04. In 12.04 the same error message appeared after waking from sleep mode, but then another followed that the tablet was activated again and it worked properly. Any ideas how this could be fixed? This issue happens only on UEFI systems with enabled Secure Boot.  When I try to install DKMS modules like VirtualBox, Nvidia, or Broadcom drivers, they do not install and I get Required key not available when I try to modprobe them.  VirtualBox complains that vboxdrv is not loaded.  Broadcom wl driver is shown in lspci -k as a kernel module but is not in use. sudo modprobe wl throws Required key not available.  Also this issue may happen when I install some kernel modules from git sources.  This issue may appear after a kernel update as disabled wireless adapter, black screen after a reboot, etc.   How can I fix it? List.Add seems to be duplicating entries. What's wrong? javax.persistence.PersistenceException : [PersistenceUnit: vodPersistenceUnit] class or package not found "We plan" versus "we are planning" Is there a semantic difference between the following two sentences?     In the future, we are planning to migrate our tool to the Z3 solver.         In the future, we plan to migrate our tool to the Z3 solver. given the 2 PDE  $$ \Delta u-au_{tt}+u_{t}=0$$  and $$ \Delta u + Du*Df=0 $$  here $ \Delta u $ is the Laplacian $ Du= grad(u) $ is the gradient and * means scalar product $u_{t} = \frac{\partial u}{\partial t}$  my doubt is what term should i include to get the linear part of the equations $ u_{t} $ and $Du*Df $ from the Euler Lagrange equations in $ R^{n} $ thanks How can I make a view that seems like orthographic, but not exactly? For the start, I thought was orthographic, but when I used the orthographic camera, it didn't match up. Client say they want a so-called &quot;industry schematic&quot;, talking about a 135 degree look.  I found a trick to do this, which is using lattice. I align the lattice with the cube, and move the upper four points a little bit along the axis showing on the following image. But I want to achieve this by just using the camera. How to expand the ext4 primary partition size in a VMware Player virtual disk? I have a VMware Player virtual machine setup, running Ubuntu 10.04 as the a guest OS.  The VM was created on a 20GB virtual drive, with a single partition for working space. Finally I got to the point where I need to expand the size of the virtual drive. Resizing the drive itself, adding some 10GB, was easy via the VMware Player menus. I now need to resize the main partition itself (I prefer this over creating a separate partition and mounting it).  According to GParted, I now have the following partitions:    My question is, how can I increase the size of the Filesystem partition to include the free 10GB?  From information I found online, it looks like one limitation is the fact that he swap partition is immediately adjacent to the filesystem, so my understanding is that first thing would be to move that partition to the end of the volume. How can this be done?  Also, some sources suggest that the system should be booted from a LiveCD, so the filesystem is not the boot partition, and thus make it easier to resize. As mentioned above, I run a VM, so I am not sure how to boot it from a LiveCD. Any ideas? How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$? How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$?  If $\varphi$ is one of this automorphism then $\varphi((1,0))=(1,0),(1,3),(3,3),(3,0)$ and $\varphi((0,1))=(0,1),(0,5),(2,1),(2,5)$.  So it seems to me that this groups contains 16 elements, am i right? And if I'm right there is a way to understand to which group of cardinality 16 this is isomorphic to?  EDIT: $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}\cong\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$. And $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ are characteristic (am I right?), so $\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z})\cong\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z})\times\mathrm{Aut}(\mathbb{Z}/3\mathbb{Z})\cong\mathbb{Z}/8\mathbb{Z}^*\times\mathbb{Z}/3\mathbb{Z}^*$ and this group has cardinality 8, right? Where is my mistake? Flag and close my own question Why is it possible to flag and close my own question?   The only realistic usage I could think of is when I see that my question is a duplicate, but it has already got some great answers. So instead of deleting the question, I would vote to close it as a duplicate.   But still, I can't see how flagging my own question would be useful. What is the basic concept behind hooks? I am intermediate in PHP. To polish my skills I start learning Drupal 7. While learning Drupal architecture concepts, the terms hooks and bootstrapping puzzled me a lot. I read the "Pro Drupal development" book, and some documentation on drupal.org, but it is so advanced for me to learn how hooks work in Drupal to display the web page.  Can anyone tell me what hooks are in simple words? How do I reset my manor so I can rebuild it with different options? I currently have Windstad Manor and built it all. Now I regret some choices and I'd like to rethink and rebuild it.  Is there any console way to "nuke" the manor and rebuild it from ground up?