A Linear map A whose trace is 0. I want to know how to prove that there exists a group of base . these base make the matrix of this map is a matrix whose Principal diagonal element is all zero.
If $\mathrm{Tr}(A)=0$ then $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$ Prove that if $A$ is a square matrix and $\mathrm{Tr}(A)=0$, then there exists an invertible matrix $R$ such that the matrix $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$.   It seems like the formula $A=S\Lambda S^{-1}$, but maybe it does not help.  Thanks so much.
How is the width of a slit related to the intensity of light passing through it? Here's a question I got in my final exam this morning. &quot;If in Young's double-slit experiment setup, the ratio of the intensity of the bright spot to the dark spot is 25:9, what is the ratio of the width of the slits?&quot; Here's what I did. Since the ratio of intensity at the bright and dark spots is 25:9, the ratio of amplitudes there must 5:3. Which means the amplitude of one wave is 4 times the other. Now, knowing that the amplitude of light through the wider slit is 4 times the amplitude of light through the narrower slit, how can I determine the ratio of the slits' width? Why was a regular train used to transport students to Hogwarts? How to Self-Study Mathematical Methods? Edit:  Ok, user  made comment that truly helps narrow the focus of my question. Basically, I'm asking for a self-study course of Mathematical Methods.  Thanks to his recommendation I identified a book that is very close to the answer:  Mathematical Methods in the Physical Sciences by Mary L. Boas  Supposedly, the book covers most of the Math a person needs to learn in order to understand the simulations I presented.  Does this book suit my needs?    Hello,  I have always been interested on the idea to study Math at an Undergraduate level to be able to implement simulations like the followings:  Finite Difference Simulation of the Wave Equation    Heat Equation in Two Dimensions    Besides my precollege experience in Math, I have only been able to do very simple simulations due to my lack of knowledge of Advanced Math. Every time I ask about the inner workings of the simulations I get redirected to papers that look like foreign language to me and, unfortunately, I depend on easy-to-read tutorials to understand them, and while I have found   I'm more interested in reaching the solution myself.  I checked some of the questions related to book recommendations for self-study and while they are very helpful I'm interested in learning by example (There is a book called: "Accelerated C++ Practical Programming by Example" that fits very well my study habits)  Thanks in advance. Area of an irregular polygon I want to show that if a real orthogonal matrix $A$ has determinant $-1$ then $\lambda=-1$ must be an eigenvalue of $A$.  I have proven this in a long-winded way and I was wondering if these is a quick way of seeing it. Interior product (contraction) symbol See the above excerpt from John Lee's Introduction to Smooth Manifolds. In differential topology and linear algebra, there's an operator called the interior product or contraction operator, denoted $\iota_X(\omega)$ or alternatively with some other "half box" like operator $X_|\omega$. Does anyone know how to markup that operator in LaTeX?   Google has turned up nothing except for this thread:   That thread did receive an accepted answer, which was \invneg from the MnSymbol package, but the suggestion given there does not seem correct to me. Or at least, it does not match what I'm familiar with. That answer results in an inverted negation symbol, whereas the contraction operator should have a stroke down at the floor of the line, and a long stroke going up almost the full height of the line. It can be found in many texts, but why can I not find the correct LaTeX markup? Prove previsibility and $E[X_{T \wedge n}] \le E[X_{S \wedge n}]$ Suppose I want to know whether $\sqrt{7}\in\mathbb{Q}_5$, or more generally, whether $\sqrt{n}\in\mathbb{Q}_p$ for $n\in\mathbb{Z}$, $p$ an odd prime. What are the techniques for determining this? Am I supposed to "lift" it from $\mathbb{Z}/p\mathbb{Z}$ using Hensel's lemma? Could I see an example? How to host my existing website with Google Drive Google has given me 15GB of storage. Can I host my own website having my own domain? How do I run a game for a larger group? I am currently running a Dungeons and Dragons 4 campaign for a group of 9 players.  Whittling the group down to a more manageable size is not really an option since they are all family.  When we do combat it takes forever even if I have one player help me and allow them to go 2 at a time.  When we do story based adventure there are 2-3 players that do most of the talking and the others twiddle their thumbs.  I've tried encounters with many foes, encounters with only a few tough foes, dialog only encounters, social puzzle encounters, everything I can think of.  What else can I try to get this group more engaged? What do you call someone who acts and thinks it is helpful, even though they do not know the context of what is going on? What is a name for a person who assumes they are doing something helpful, even though they are ignorant to the situation? Someone who reacts without bothering to understand what’s going on, and feels as though they are doing good? Is there such a word?   For instance, someone who pulls another out of the pool, and pats them self on the back for saving them from drowning...without ever asking if the person was drowning or merely swimming. Say I have a Zombie token, , and  on the battlefield. Can I sacrifice the Zombie token to activate both the Altar's and Seer's abilities?  Similarly, if I have one card in hand, and  and  on the battlefield can I discard the one card to both of their abilities?  If I have a  with 3 charge counters can I activate both of its abilities by tapping it to add a counter, and tapping it to remove 3 counters and destroy a permanent without untapping it in between? Create ColorRamp How can I create a ColorRamp for a Cycles material through Python?  I am able to create a new shader output node with nodes.new(type='ShaderNodeOutputMaterial'). Searching online, I see that it is only possible through C code (however, these pages dates back several years).  Is there a way to fake a ColorRamp (maybe with multiple Math nodes in greater/less than mode) or create a ColorRamp for a Cycles material through Python? My options to link large image into AI file My .psd file has a resolution of 6m x 3m @ 150 DPI. This makes it 35433px x 5906px in size. Photoshop CS6 saves the file as .tiff or .psb only. When I first import into Illustrator I get an error that the file is not readable but I have tried many options. Last one is not supported by AI.  So I'm puzzled... is there a way to link large images into Illustrator? Why is this regex allowing a caret? Table fixed header and scrollable body How can I access Apache (on VirtualBox guest) from host? I installed Apache on an Ubuntu VM. When I go into the guest and load up Firefox I can see that Apache is working fine when I browse to localhost.  I would like to access this same page through the host. I've tried using the IP address of the guest but nothing comes up. Ideally I would love to set up my Windows 7 (host) to request http://guestserver.com of the Apache server on the guest. How can I do this? System.ListException: Before Insert or Upsert list must not have two identically equal elements - Deep Dive How do I access a specific element in a vector of vectors clojure I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$  I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively things like $${2n+2\choose n+1}={2n+2\over n+1}{2n+1\choose n}=2\cdot {2n+1\over n+1}{2n\choose n},$$  But I don't think it can lead me anywhere. I would like the proof to be as simple as possible.