If you have a symmetric matrix, is it orthogonally diagonalizable? Or is the converse only true?
Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = Q\Lambda Q^{-1} = Q\Lambda Q^{T}$ where $\Lambda$ is diagonal.
Graphics issues after/while installing Ubuntu 16.04/16.10 with NVIDIA graphics I am having some issues when rendering an image, my rendering screen is turning into grey even when making a full render. What am I doing wrong? What is the typographic reasoning here? Does it make the text look less crowded? How do you start Windows 10 in safe mode? I've been through the Windows 10 upgrade, and am experiencing a recurring "INTERNAL_POWER_ERROR", which is stopping me from logging in.  I'd like to be able to try to boot to safe mode, but it appears that pressing F8 during boot is no longer enough, so how can I boot in safe mode (or any of the other recovery options) on a Windows 10 PC that doesn't let me log on? Add default syntax highlighting for the [dart] tag I just approved several edits from a user adding  to questions with the  tag. Apparently , Google Code Prettify has added support for Dart.  I suggest it's time to follow suit and update the default syntax highlighting associated with the  tag. Adding it manually to every question sounds like way too much work, even if I all I have to do is click the "Approve" button. How do I get my Mac to boot from an Ubuntu USB key? Parsing input with scanf in C I've been having a lot of problems trying to figure out how to use scanf().  It seems to work fine with integers, being fairly straight forward scanf("%d", &amp;i).  Where I am running into issues is using scanf() in loops trying to read input.  For example:  do {   printf("counter: %d: ", counter);   scanf("%c %c%d", &amp;command, &amp;prefix, &amp;input); } while (command != 'q');    When I enter in a validly structured input like c P101, it seems to loop again before prompting me.  This seems to happen even with a single:  scanf("%c", &amp;c)    in a while loop.  It'll do the loop twice before prompting me again.  What is making it loop twice, and how do I stop it? When I enter in less amount of input that programmatically wouldn't have another character or number such as q, pressing enter seems to prompt me to enter more.  How do I get scanf() to process both single and double character entries? Is there a way to bypass resistance/immunity to fire damage? I want to use fire-based spells for thematic reasons and a lot of things have immunity to it.   Is there a feat or class feature I could use to bypass/reduce those? For example, is there something that ignores fire resistance and reduces fire immunity to just resistance?   I know the easy way around it is just to pick another damage type spell but I really want to roleplay a pyromancer/firebender-type character. I am just curious that the following format looks good to you or not? The preamble I used is just   \documentclass[12pt,a4paper]{article}  \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb}   I don't feel it's so good because the first paragraph has no indent but the second does. If the first paragraph is much longer, than it's fine. But if not, it looks strange. Should I leave it, or you have suggestions for this? Algebra of LDA. Fisher discrimination power of a variable and Linear Discriminant Analysis Apparently,     the Fisher analysis aims at simultaneously maximising the   between-class separation, while minimising the within-class   dispersion. A useful measure of the discrimination power of a variable   is hence given by the diagonal quantity: $B_{ii}/W_{ii}$.     I understand that the size (p x p) of the Between (B) and Within-Class (W) matrices are given by the number of input variables, p. Given this, how can $B_{ii}/W_{ii}$ be a "useful measure of the discrimination power" of a single variable? At least two variables are required to construct the matrices B and W, so the respective traces would represent more than one variable.  Update: Am I right in thinking that $B_{ii}/W_{ii}$ is not a trace over a trace, where the sum is implied, but the matrix element $B_{ii}$ divided by $W_{ii}$? Currently that is the only way I can reconcile the expression with the concept. How to tail multiple files using tail -0f in Linux/AIX How do you read C declarations? When can we speak of collinearity Is it possible to tell when statistics were last accessed by a query using T-SQL? I'm trying to ascertain what statistics are involved in a query, I know I can check the visual query plan but I wondered whether there was a table (or tables) with this information. If $A$ is a square matrix and $A^2 = 0$ then $A=0$. Is this true? If not, provide a counter-example. How to hide a folder in sdcard in android In the Android App, if a question is closed, I won't be able to leave comments on either the question or any of its answers. There are simply no "Add Comment" buttons available anywhere if the question is closed.  See the screenshot below. The comment adding interface is simply removed on closed questions and their answers &mdash; no Reply button in the toolbar, and no Add Comment button under the question.        Occurs on Android app v1.0.50 and v1.0.51, on a HTC One running Android 4.4.3. Why is the decay of a neutral rho meson into two neutral pions forbidden? I have a simple circuit:    The max rating (current) of the LED is 30mA.  How can I work out what the resistance of the resistor needs to be? By using ohm's law I found it to be \$3V/0.03A = 100 \Omega\$. However using software called , and trial and error I got the minimum possible resistance to be 36&Omega;. However, if I use a 35&Omega; resistor, then the LED breaks. Is the software wrong, or (more likely) am I doing something wrong? Multiple TikZ keys with foreach