Show that: If $X \in \mathbb{R}^{2x2}$ is a matrix which satisfies the relation $X \cdot Y = Y \cdot X$ for all $Y \in \mathbb{R}^{2x2}$, then $X$ has the form $X = \begin{pmatrix} \lambda &amp; 0\\  0 &amp; \lambda \end{pmatrix}$ with $\lambda \in \mathbb{R}$.    I don't know how to solve this task? Maybe we are allowed to set specific matrix for $Y$? But I'm not sure because it's saying "for all" so we must stay general?  Else I would say we can satisfy this relation when we set $Y = \begin{pmatrix} 1 &amp; 0\\  0 &amp; 0 \end{pmatrix}$ I think.   But that's not enough to prove it right? : /
Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \  \lambda \in \mathbb R\}$$  ($I_n$ is the $n \times n$ identity matrix)  I don't know how to use $E_{ij}$ (matrix with $1$ in $(i,j)$ and $0$ elsewhere) and the elementary matrix $P_{ij}$ to prove this question. Can anyone explain it please?
Non-root login on EC2 instance For a given ellipse, find the locus of all points P for which the two tangents are perpendicular.   I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry proof. I read it as a paperback around 2009ish. There are at least two books in the series. In the first book, we learn that the protagonist lives up in the mountains. Her people are distrusted by the people in the plains, and I think have either dark or olive skin. She leaves home to learn magic, something I think is rare in the world she lives in. When she arrives, there's a crazy power-hungry man at her destination and she has to hide who she is because otherwise he'd have her killed to prevent her from surpassing him. At some point, she winds up in a set of chambers set into the mountain where she finds a pool of water and carvings about dragons. Somehow, this leads into her learning to become a dragon, I think perhaps after meeting one. I want to say that the sequel got into her having dreams that turn out to be partly prophetic and partly communication.  One of the reasons the book sticks out in my memory is that, despite the mentions of the protagonists dusky skin, the cover depicts a fair-skinned blonde girl, I think holding a glowing globe of light. What GUI applications are there to assist in generating graphics for TeX? I found  to be a useful tool to help generate code for graphs. I suppose there must be other tools to help draw diagrams such as Sets/Venn Diagrams or Electrical Circuits (Logic Gates). Which ones would you suggest? Calling Python in PHP I have a Python script I recently wrote that I call using the command line with some options. I now want a very thin web interface to call this script locally on my Mac.  I don't want to go through the minor trouble of installing mod_python or mod_wsgi on my Mac, so I was just going to do a system() or popen() from PHP to call the Python script.  Any better ideas? Thanks in advance! Why is fundamental group insufficient to classify manifolds? I have heard that some issues in group theory prevent classifying all manifolds upto homotopy using the fundamental group invariant. Does anyone know what are those issues?  Thanks, K. Initializing a member array in constructor initializer Meaning of $\int\mathop{}\!\mathrm{d}^4x$ What the following formula mean?  $$\int\mathop{}\!\mathrm{d}^4x$$  I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following $\int\mathop{}\!\mathrm{d}^4x$ leave empty the argument and also have the $\mathrm{d}$ elevated to the fourth power. What that mean?  Update.  In the Einstein-Hilbert action we have (note that I have understand that the other parts of the integral are relevant only after your answers, thus sorry.):  $$S=\frac{c^4}{16\pi G}\int\mathop{}\!\mathrm{d}^4x \, \ R \sqrt{-g}$$ Can open, unsafe nuclear fusion reaction burn the atmosphere? I happened to hear people saying that the nuclear fusion bomb tests could set the atmosphere on fire. I have some serious doubts about that - but I have no facts. Nuclear fusion reaction requires $15*10^{6}$ kelvins to start. If we produce such temperature in "open air" would the atmosphere become a fuel for further fusion? Shouldn't the whole thing just be torn apart by its terrible pressure? Proving that a function has a removable singularity at infinity If I run along the aisle of a bus traveling at (almost) the speed of light, can I travel faster than the speed of light? FME - Calculate distance of point along a line I have created an FME workspace which inputs a pipeline as well as other linear features which cross it (waterways, roads, railways, fences, other pipelines, etc).  The output is a point file which contains the crossing type.  What I would like to do is calculate the distance along the pipe from the start of the pipeline for each intersection.  So, as you travel along the pipeline, the distance attribute will increase.   I'm thinking that I'll need to use some of the linear referencing tools like MeasureGenerator, MeasureExtractor, or something like that, but I'm not sure how to do so.  Does anyone have an idea of how to do this? Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups I have a finite $p$-group $G$ and a normal subgroup $N$ which is not the trivial subgroup. I am asked to show that $|N \cap Z(G)| &gt; 1$.  There has been a similar question on MSE here:  Unfortunately, I cannot comment on other user's questions yet, so I made my own question.  I know that the trick is to use conjugation but I can't follow the logic. Can someone elaborate? Especially on the last part of the top answer. Python Flask shutdown event handler Solutions to $a+b+c=12$, $a, b, c \in \mathbb{N}_0$ Let $a, b, c \in \mathbb{N}_0$. If $a+b+c=12$, how many solutions $(a,b,c)$ satisfy the equation?   Is the answer:      $729$ Apply empty style to the entire bibliography I installed the VMware bundle on my Ubuntu 11.04 successfully but when I open it it gives me this window     and I don't know the path to this C headers. How to list the tables in a SQLite database file that was opened with ATTACH? What SQL can be used to list the tables, and the rows within those tables in an SQLite database file - once I have attached it with the ATTACH command on the SQLite 3 command line tool? After discussion with colleagues regarding the use of the 'var' keyword in C# 3 I wondered what people's opinions were on the appropriate uses of type inference via var?  For example I rather lazily used var in questionable circumstances, e.g.:-  foreach(var item in someList) { // ... } // Type of 'item' not clear. var something = someObject.SomeProperty; // Type of 'something' not clear. var something = someMethod(); // Type of 'something' not clear.   More legitimate uses of var are as follows:-  var l = new List&lt;string&gt;(); // Obvious what l will be. var s = new SomeClass(); // Obvious what s will be.   Interestingly LINQ seems to be a bit of a grey area, e.g.:-  var results = from r in dataContext.SomeTable               select r; // Not *entirely clear* what results will be here.   It's clear what results will be in that it will be a type which implements IEnumerable, however it isn't entirely obvious in the same way a var declaring a new object is.  It's even worse when it comes to LINQ to objects, e.g.:-  var results = from item in someList               where item != 3               select item;   This is no better than the equivilent foreach(var item in someList) { // ... } equivilent.  There is a real concern about type safety here - for example if we were to place the results of that query into an overloaded method that accepted IEnumerable&lt;int&gt; and IEnumerable&lt;double&gt; the caller might inadvertently pass in the wrong type.  var does maintain strong typing but the question is really whether it's dangerous for the type to not be immediately apparent on definition, something which is magnified when overloads mean compiler errors might not be issued when you unintentionally pass the wrong type to a method. Normally, LaTeX will only create references for the BibTeX entries cited in the text. Is there a way of extracting these entries into a different .bib file automatically? For example if I have a .bib file with two entries, and only one is cited in a particular text, I need a way of creating a new .bib file with just that reference.