I need some help at the following exercise:  "Prove that if $ A,B \in M_{n \times n} ( \mathbb{R})$, then the matrices $AB$ and $BA$ have the same eigenvalues."   Could you give some hint?  I thought that I could use the fact that:  If A and B are two square n×n matrices then characteristic polynomials of AB and BA coincide.  So $det(AB- λ I)=det(BA- λ I).$ How could I continue? Or is this way wrong?
Let $A$ and $B$ be $m \times n$ and $n \times m$ real matrices.  I proved that if $\lambda$ is a nonzero eigenvalue of the $m \times m$ matrix $AB$ then it is also an eigenvalue of the $n \times n$ matrix $BA$. But I am having trouble finding an example showing that this need not be true if $\lambda=0$  Thank you in advance
How can I automate the workflow for producing multiple versions of a document? Say you have two versions of the same LaTeX document.    For instance, you have one version of your lecture slides for actually delivering a lecture: as you're really not a big fan of the "wall of text", this version would be parsimonious in terms of text content, allowing the audience to listen to what you're talking  about instead of reading the slides, and giving graphical content centre stage.  Another version of those slides, made available to the audience after the lecture, for self-study, would be more self-contained and include a summary of what you said during that lecture along with the graphical content mentioned above.  Consider that you can toggle between the two versions by either changing a class option or by changing the value of some dedicated counter in your input file; something relatively simple to do "by hand".  To produce two different versions of the same document, my current workflow is   "manually" compile the first version, change the filename of the output produced (in order to prevent overwriting in step 4) make appropriate changes in my input file to switch to the second version, "manually" compile the second version.   However, this workflow is tedious and error-prone, especially when the number of different versions increases. I would like to automate it as much as possible. Is there a way to batch compile all the versions of my document, each getting a different output filename?  I guess that would involve modifying the input file at the command line (I'm on Mac OS X), but I don't know how to do that.  Do you have an automated workflow for compiling multiple versions of the same document? How do you do it?  Edit: I figured it out using  and  on , of which my question is a duplicate. Should we `enable` the User Systemd service `pulseaudio.service`? Air gap on isolation space The below photo belongs to a PSU. I wonder what was the reason of placing air gaps on the 8 mm isolation. Is it to improve the isolation or provide a thermal relief (although there is no component below the red circled air gap)? I tried to make a crouch system. In my script I changed .height and .center properties of collider in CharacterController component. But there are some problems. First, once I press "crouch button" collider is changing its height and center position but when I release the key the collider isn't returning to its previous state. Secondly, if I change its height and position by hand when collider is under a block, it's just stuck. How can I fix this problem?   Here's my script:  using System.Collections; using System.Collections.Generic; using UnityEngine;  [RequireComponent(typeof(CharacterController))] public class PlayerController : MonoBehaviour {      public float moveSpeed;     public float jumpForce;     public CharacterController controller;     public GameObject myCamera;      private Vector3 moveDirection = Vector3.zero;     public float gravityScale;      public Animator anim;      // Use this for initialization     void Start () {         transform.tag = "Player";         controller = GetComponent&lt;CharacterController&gt;();            }      // Update is called once per frame     void Update () {         Vector3 forwardDirection = new Vector3 (myCamera.transform.forward.x, 0, myCamera.transform.forward.z);         Vector3 sideDirection = new Vector3(myCamera.transform.right.x, 0, myCamera.transform.right.z);         forwardDirection.Normalize ();         sideDirection.Normalize ();         forwardDirection = forwardDirection * Input.GetAxis("Vertical");         sideDirection = sideDirection * Input.GetAxis("Horizontal");         Vector3 directFinal = forwardDirection + sideDirection;          if (directFinal.sqrMagnitude &gt; 1)         {             directFinal.Normalize ();         }          if (controller.isGrounded)         {             moveDirection = new Vector3(directFinal.x, 0, directFinal.z);             moveDirection *= moveSpeed;             if (Input.GetButtonDown("Jump"))             {                 moveDirection.y = jumpForce;             }              if(Input.GetButtonDown("Crouch"))             {                 controller.height = 1f;                 controller.center = new Vector3(0f, -0.5f, 0f);                 moveSpeed = 3f;             }         }         moveDirection.y = moveDirection.y + (Physics.gravity.y * gravityScale * Time.deltaTime);         controller.Move(moveDirection * Time.deltaTime);          anim.SetBool("isGrounded", controller.isGrounded);         anim.SetFloat("Speed", (Mathf.Abs(Input.GetAxis("Vertical")) + Mathf.Abs(Input.GetAxis("Horizontal"))));     } }   Thanks for any help! What is the name for moments, a parameter that is calculated What's the simplest way to do a find and replace for a given input string, say abc, and replace with another string, say XYZ in file /tmp/file.txt?  I am writting an app and using IronPython to execute commands through SSH &mdash; but I don't know Unix that well and don't know what to look for.  I have heard that Bash, apart from being a command line interface, can be a very powerful scripting language. So, if this is true, I assume you can perform actions like these.  Can I do it with bash, and what's the simplest (one line) script to achieve my goal? I would like to redirect a URL with uppercase characters http://localhost/A/B to a lowercase version http://localhost/a/b using the .htaccess file. Using regex I can capture A and B but is it possible to convert them to lowercase? Clicking on a notification on Android app opens an empty feed When I receive a message on an SE post, I get a notification like this:    If I double tap this notification, the SE application opens, but my feed is empty approximately half the time. It looks like this:    If I refresh the feed (slide from the top), the feed does refresh and shows the new notification.   The notification, above, leads to  comment.  I haven't figured out the pattern yet, but this occurs roughly half the time I open a notification. It's occurred for chat messages, comments and answers - but not consistently. I have received notifications for the same post (for example, the two comments after the one linked above) that do not show an empty feed. But, a chat message barely 10 minutes later will show the empty feed again.  I am utilizing the beta app - version 1.0.72 on Android 5.0.1. Why is it usually "friend of his", but no possessive apostrophe with "friend of Peter"? As  shows, we nearly always use the possessive form of personal pronouns for friend of mine/his/ours/etc.  But when it comes to actual names, we prefer  without the possessive apostrophe. That preference is even more marked with, say, . Not that I think the usage itself is particularly American - it's much the same with .  Personally, I find friend of him grates. In general I've no strong feelings either way as to whether it's friend of Peter or friend of Peter's (though I deplore the possessive in ), but in line with many others, I really don't like the possessive in relation to things like countries.  Why is this?   Edit: Noting an apparent "progression" (pronoun -> person -> nation) marked by reduction in use of the possessive, I checked at a finer "granularity". NGram shows that although it does occur,  virtually "flatlines" against friend of mine. But the bias reduces through of you, of us, and by the time I get to  it's much less extreme. There seems to be something "egocentric" about the double possessive.  Presumably when babies learn to speak, they soon notice that possessive pronouns, possessive apostrophes, and the word "of", all do the same job. Parents would correct a child who says "of mine's", but probably wouldn't even notice the same "redundancy" in "of Peter's". Younger speakers are unlikely to even be talking about something "of America's". Perhaps as we mature we tend to discard the "double possessive" for the more "distant" things that only adults are likely talk about, but we keep it for "closer" people because that's how we spoke when we were younger.  EDIT2 I note that I'm a great fan of him is vanishingly rare compared to ...fan of his, but with ...fan (of John) the double possessive occurs far less often than ...friend (of John's). Usage seems to be affected by the noun before "of" as well as the one after it. This is getting complicated... Infinite series for $e$... How do you prove that $e=\sum_{n=0}^{\infty}\frac{1}{n!}$? Here I am assuming $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$. Do you have any good PDF file or booklet available online on this? I do not like how my analysis text handles this... I noticed that after joining a table to a shapefile the headers of each fields in the table used for the join are changed by QGIS and they are attributed to a different name.  For instance, I had a shapefile with vegetation_codes and a table with vegetation_codes and veg_definition (these were the headers). After joining the table to the shapefile and also saving the shapefile and creating a new one, the vegetation_classes header was changed into vegetati_1.  Is there a way to keep the original headers?  I am using QGIS Dufour on Windows 7 and the table that I join to the shapefile is CSV (tried with dbf as well and the problem is the same). Finding an isomorphic subring of matrices I'm struggling a fair amount with this exercise:     Find a subring of $M(2,\mathbb{Q})$ which is isomorphic to a) $\mathbb{Q}$ x $ \mathbb{Q}$ b) $\mathbb{Q}$ c) $\mathbb{Q}[x]$/$x^2$   Now I know a subring must be a subgroup, must contain the elements $0,1$ and must be closed under multiplication. As we are looking for isomorphisms then they must also be ring homomorphisms and must be bijective.  I tried to come up with a random subring and attempt to prove it is an isomorphism. E.g $\left\{\left.\begin{matrix}  a &amp; b \\ c &amp; 0  \end{matrix}\right|a,b,c \in \mathbb{Q}\right\}$ is a subgroup of $M(2,\mathbb{Q})$ and then try and show it's an isomorphism. However I'm stuck here as I don't actually know how to go about finding these specific isomorphisms and I completely lose what I'm doing. Any help would be great. A geometry question: Let $ABCD$ be a convex quadrilateral. Prove that AD is parallel to BC. Let $ABCD$ be a convex quadrilateral such that the length of the segment connecting midpoints of the two opposite sides $AB$ and $CD$ equals $\frac{AD+BC}{2}$ . Prove that AD is parallel to BC. I assume that AD is not parallel to BC but I can't find any contradiction. Question is :     Prove that the polynomial $p(x)=x^4-4x^2+8x+2$ is irreducible over the quadratic field $F=\mathbb{Q}(\sqrt{-2})$. [Hint : first use the method of proposition $11$ for the U.F.D $\mathbb{Z}[\sqrt{-2}]$(cf.Exercise $8$, Section $8.1$) to show that if $\alpha \in \mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)$ then $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$ . Conclude that $\alpha$ must be $\pm 1,\pm \sqrt{-2}$ or $\pm 2$ and hence show that $p(x)$ has no linear factor over $F$. Show similarly that $p(x)$ is not the product of quadratics with coefficients in $F$.]   What I have done so far is :  Suppose $\alpha \in\mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)=x^4-4x^2+8x+2$ we would then have :  $p(\alpha)=\alpha^4-4\alpha^2+8\alpha+2=0\Rightarrow 2=\alpha(-\alpha^3+4\alpha-8)$  i.e., $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$.  so, I have used the method of proposition $11$ for the U.F.D $\mathbb{Z}[\sqrt{-2}]$(cf.Exercise $8$, Section $8.1$) to show that if $\alpha \in \mathbb{Z}[\sqrt{-2}]$ is a root of $p(x)$ then $\alpha$ is a divisor of $2$ in $\mathbb{Z}[\sqrt{-2}]$  I do not understand why does he mentioned that $\mathbb{Z}[\sqrt{-2}]$ is U.F.D and all... I do not use that at all... It is unnecessarily confusing me or i am unnecessarily getting confused.. hint is actually misleading me :(  Now, I have to prove that $\alpha$ must be $\pm 1,\pm \sqrt{-2}$ or $\pm 2$  i.e., suppose I have $2=ab$ in $\mathbb{Z}[\sqrt{-2}]$ then, $N(2)=N(ab)\Rightarrow 4=N(a)N(b)$   i.e.,$N(a)=1\text{ or }2\text{ or } 4$ i.e., $p^2+2q^2=1\text{ or }2\text{ or } 4$ for $a=p+\sqrt{-2}q$   $p^2+2q^2=1\Rightarrow p=\pm 1 \Rightarrow a=\pm 1$ $p^2+2q^2=2\Rightarrow q=\pm 1\Rightarrow a=\pm\sqrt{-2}$  $p^2+2q^2=4\Rightarrow p=\pm 2\Rightarrow a=\pm 2$    Once I prove that those are the only divisors then I would consider :    $p(1)=(1)^4-4(1)^2+8(1)+2\neq 0$ $p(-1)=(-1)^4-4(-1)^2+8(-1)+2\neq 0$ $p(\sqrt{-2})=(\sqrt{-2})^4-4(\sqrt{-2})^2+8(\sqrt{-2})+2\neq 0$ $p(-\sqrt{-2})=(-\sqrt{-2})^4-4(-\sqrt{-2})^2+8(-\sqrt{-2})+2\neq 0$ $p(2)=(2)^4-4(2)^2+8(2)+2\neq 0$ $p(-2)=(-2)^4-4(-2)^2+8(-2)+2\neq 0$   So, no divisor of $2$ is a root..  Thus $p(x)$ do not have a root in $\mathbb{Z}[\sqrt{-2}]$  suppose I have something like :  $$x^4-4x^2+8x+2=(x^2+ax+b)(x^2+cx+d)=x^4+(a+c)x^3+(b+d+ac)x^2+(ad+bc)x+bd$$  But then, $bd=2\Rightarrow \text { b,d  are  a divisors of 2 in $\mathbb{Z}\sqrt{-2}$}$  But then we have seen that only divisors of $2$ in $\mathbb{Z}\sqrt{-2}$ are $\pm 1,\pm \sqrt{-2}$ or $\pm 2$  So, only possibilities are   $x^4-4x^2+8x+2=(x^2+ax\pm 1)(x^2+cx\mp 2)$ $x^4-4x^2+8x+2=(x^2+ax\pm \sqrt{-2})(x^2+cx\mp \sqrt{-2})$   But these are not possible...  This only tells that $p(x)$ is irreducible in $\mathbb{Z}[\sqrt{-2}]$ but then how do i show this is irreducible in $\mathbb{Q}(\sqrt{-2})$ I was expecting gauss lemma to help but it only works for integers and rationals...  So, please help me to clear this..  Thank you...   P.S : Proposition $11$ is Rational root theorem and Exercise $8$ is that ring of integers of $\mathbb{Q}(\sqrt{-2})$ is an Euclidean domain. Is infinity a number? Is infinity a number? Why or why not?  Some commentary: I've found that this is an incredibly simple question to ask &#8212; where I grew up, it was a popular argument starter in elementary school &#8212; but a difficult one to answer in an intelligent manner. I'm hoping to see a combination of strong citations and solid reasoning in the answers. Reputation widget failure on Android The title says it all. When I try to place the reputation widget on my home screen, I get this:  When I try again, I get this error, presumably because the Stack Exchange service has stopped.  Is this a bug? If so, where should I report it?  I am running Android 5.1 armeabi on a DigiLand DL718M tablet. Lowering elevation of lake using QGIS I am in QGIS. I have a raster layer of mountains and a lake. I have a vector layer (shapefile) of the lake. It sits nicely where it supposed to be over the raster level. I am going to convert this height map to an STL so that I can cnc this out of wood, so I'd like the actual lake to be something like 200 feet lower than it shows on the raster layer so that I can fill it with blue epoxy. I have read a lot of clipping and everything else, and I am lost. Effect Of Frequency on Photocurrent The situation is this.. I am performing a photoelectric effect experiment - just the usual - evacuated glass tube containing two metal electrodes connected to an external source of DC voltage (say an alkaline cell) and ammeter in series. I shine a light on one of the electrodes - the frequency is more than enough to cause eject photoelectrons, and the voltage of the cell opposes the electrons which are trying to get to the other electrode. The question is this - what happens to the current on increasing the intensity of light while keeping no. of photons falling on the electrode constant (in other words increasing the frequency)?   Does the current remain the same because the no. of photons falling on the electrode is constant and so no. of electrons being ejected is constant? Or does it increase because the electrons have more kinetic energy, ie, they are moving faster towards the other electrode so that the $t$ (time) term in $I$ = $q$/$t$ is less ($q$ remains constant because no. of electrons being ejected is still the same)?   This question appeared in one of my tests and the answer key says that the current remains constant but think that case 2 is correct and current would increase, which is why I asked this question.  Thanks. I'm trying to watch for any new output of a log file. Another script (not under my control) is deleting the file then creating a new one with the same name. Using tail -f doesn't work because the file is being deleted. Multiple Image in Canvas ItemsControl