we can turn every triangular matrix to a diagonal matrix with elementary Matrices. So; How can it possible to have a matrix which is similar to a triangular matrix but not to a diagonal matrix.    in addition, what is the best trick to know if a matrix is diagonalizable or not?
I hope it is alright to ask something like this here, I am having trouble keeping up with all the special cases and my book is being kind of vague.  I know how to do the standard method of finding diagonal matrices, but I know that triangular matrices are special and my book makes a connection, but it is hard to tell what it is.  Thanks.
linear transformation $T$ such that $TS = ST$ Let $V$ be a finite-dimensional vector space over $F$. Let $T:V \rightarrow V$ be a linear transformation such that $ST=TS$ for all linear transformations $S:V \rightarrow V$. Show that $T = cI_v$ for some $c \in F$. I have been trying to teach myself QGIS by using this forum and others to help. Most of my questions can be answered by downloading plugins that are available through 3rd party repositories but for some reason I am not able to connect with these. Anyone have any ideas why this is the case? Is it possible for test error to be lower than training error Is it possible to have test error lower than training error?  I have a classification problem with 2000 samples, 500 of which are positives, 1500 are negatives. I split my data into 70% training data, 30% test data.  Run random forest with 200 estimators and cv=10. I did this several times and compared the recall and precision score and notice the scores for my test set is significantly better. Is this possible? I would like to use client-side Javascript to perform a DNS lookup (hostname to IP address) as seen from the client's computer. Is that possible? I've been told not to use for...in with arrays in JavaScript. Why not? Search using symbols is broken We used to be able to , but that now appears to be broken.  I'd expect a  to return a result, but instead the search engine appears to put quotes around the whole request; the search box shows the search has been replaced with "[python] += list" and the  I was looking for doesn't appear among the returned questions.  Using quotes around the symbols do not seem to help here;  returns an empty result. In wikipedia FKT algorithm is given for planar graphs. Not anything for complete graphs. I need to find the number of perfect matchings in complete graph of six vertices. Why are the power transmission/distribution systems AC and not DC? Deriving of formula for finding the length of median In the below image    $AD$ is the median of $\triangle ABC$  We know that $m_A = \frac 1 2 \sqrt{2b^2 + 2c^2 - a^2} $    But can someone tell me how it's derived !! I am just unable to think of it !!   But I assume it muste be linked with the Appolonius Theorem !! Touching the front wheel with my foot when turning I'm following the , but on launching the upgrade tool I get this response:  Checking for a new ubuntu release No new release found   Am I doing something wrong? Is there a workaround? Can screen layouts or workspaces be locked? Can you add a separate discussion site? Sometimes you just want to let out some steam, and troll about some technology or software. You are not looking for any answers, but for users with similar experiences and interesting stories. So why not create a separate discussion site (trolling.stackoverflow.com?), where you could ask questions like "why do you hate technology X" or "What are the stupidest bugs you encountered in X"? Finding the limit $\lim \limits_{n \to \infty}\ (\cos \frac x 2 \cdot\cos \frac x 4\cdot \cos \frac x 8\cdots \cos \frac x {2^n}) $ I would like to see a breakdown of reputation by post type on the user summary page.    This would help show, at a glance, how much someone actually knows vs. how good they are at asking questions.  For instance:     Reputation from Questions - 10,015 from 1024 posts   Reputation from Answers - 110 from 11 posts   Reputation from Edits - 400 from 1000 edits     I would take any answers the above person gave me with a grain of salt instead of assuming they were knowledgeable because they have 10k rep.  We already have a multitude of other readouts on the new page, so this shouldn't up the clutter factor too much either. I've read before that, by the Principle of Explosion, if a theory is inconsistent, then absolutely any statement can be proven within it.  Obviously, there are statements which are independent of ZFC (Continuum Hypothesis, etc). It seems to me that this proves that ZFC is incomplete. Why does it not then follow that ZFC is consistent? It seems to me that we could say "Assume ZFC is inconsistent. Then the the Continuum Hypothesis is provable in ZFC. But the Continuum Hypothesis is neither provable nor disprovable in ZFC. Therefore ZFC is consistent."  What am I missing here? Questions with ignored tags still visible in iOS app I have set to ignore [php] tag and chose to hide questions in preferences:        But I still see questions tagged [php] in the iOS app.         Seems the app can't distinguish options for ignored tags.   App Version: 1.6.5.3 Device: iPad Air 2 (Wi-Fi) OS Version: Version 10.3.1 (Build 14E304) Can any one explain me what is the use of derivatives in real life. When and where we use derivative, i know it can be used to find rate of change but why?. My logic was in  real life most of the things we do are not linear functions and derivatives helps to make a real life functions into linear.  eg converting parabola into a linear function $x^{2}\rightarrow 2x$  but then i find this; derivation of $\sin{x}\rightarrow \cos{x}$  why we cant use $\sin{x}$ itself to solve the equation. whats the purpose of using its derivative $\cos{x}$. Please forgive me if i have asked a stupid questions, i want to improve my fundamentals in calculus. Aromatic rings resized with chemfig When I want to reduce the size of a molecule drawn with  I usually put a [scale=0.xx] before the argument of the command. However when I deal with aromatic rings this method doesn't work well, in fact only the carbon skeleton is resized: the 'aromaticity' is not and it's also out of center.  For example:  \chemfig[][scale=0.75]{**6(-------)}\quad   gives me:     How do I achieve a complete resize? Multiple Damage Resistance Similar questions to this have been asked, but I wasn't able to find one that answers this specifically.  Say that you have resistance to slashing damage as well as resistance to fire damage (no caveats to nonmagical damage or not).  You are fighting someone who has a flaming greataxe (just an enchanted item) that deals 1d12+STR slashing damage, plus 1d6 fire damage on hit and they hit you.    How do you calculate the damage?   (1d12+1d6+STR)/2 (1d12/2) + (1d6/2) + STR (1d12+STR/2) + (1d6/2) something else?