Let $A, B, C$ be some complex matrices. Suppose that $AB - BA = C$ and $AC = CA$. Prove that: $C^k = 0$ for some $k$.   It is an exercise in the section of "Jordan Canonical Form of Nilpotent Transforms". But I am not sure how to prove the question. Could anyone give me a hint?
I saw this in a  and am putting it here for posterity.  Problem: Let $A$ and $B$ by square matrices and set $C=AB-BA$. If $AC=CA$, prove $C$ is nilpotent.
Must a husband and wife make separate online applications for a UK Standard Visitor visa, and so attend separate appointments for biometrics? Use XeLaTeX in Texmaker instead of pdfLaTeX How many ways can $r$ nonconsecutive integers be chosen from the first $n$ integers? How to create script with auto-complete? In order to better understand the SQL Server query processor, I've been thinking about how the ORDER BY clause works, and how SQL Server provides results.  It appears SQL Server will provide results in exactly the same order for any given set of data that remains the same and never, ever changes.  Once you introduce any kind of uncertainty such as developers changing something, the order of results can no longer be expected to be the same.  Simply seeing the results in the same order, each time you press F5 does not guarantee the results you expect.  Try this:  USE TempDB; DROP TABLE TestOrderBy; DROP TABLE TestOrderBy2; CREATE TABLE TestOrderBy (     ID INT NOT NULL CONSTRAINT PK_TestOrderBy PRIMARY KEY CLUSTERED IDENTITY(1,1)     , SomeData varchar(255) ); INSERT INTO TestOrderBy (SomeData) VALUES ('E'); INSERT INTO TestOrderBy (SomeData) VALUES ('D'); INSERT INTO TestOrderBy (SomeData) VALUES ('C'); INSERT INTO TestOrderBy (SomeData) VALUES ('B'); INSERT INTO TestOrderBy (SomeData) VALUES ('A');  CREATE TABLE TestOrderBy2 (     ID INT NOT NULL CONSTRAINT PK_TestOrderBy2 PRIMARY KEY CLUSTERED IDENTITY(1,1)     , SomeData varchar(255) ); INSERT INTO TestOrderBy2 (SomeData) VALUES ('E'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('D'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('C'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('B'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('A');  SELECT SomeData  FROM TestOrderBy UNION ALL SELECT SomeData FROM TestOrderBy2;  CREATE INDEX IX_TestOrderBy_SomeData ON TestOrderBy (SomeData); CREATE INDEX IX_TestOrderBy2_SomeData ON TestOrderBy2 (SomeData);  SELECT SomeData  FROM TestOrderBy UNION ALL SELECT SomeData FROM TestOrderBy2;   The results:    As you see, adding a simple index on the fields selected in the query alters the order of the results.  From here on, I'm adding ORDER BY unless I truly don't care. I'm trying to hide the app icon from the title bar in GNOME in Ubuntu 18.04. I know there is  asking the same thing but it's from 2011 and I can't find what it requires to find in the shell css. I'm also using the Arc-Dark custom theme.   Here are two examples of what I want to remove:      To be clear, I don't want to resize it to fit, I want to straight away remove it completely. I've got a qml file and I just want to ssh to the phone and do  qmlscene myfile.qml   Is it possible without doing all that stuff with SDK and packages? Just vim and some simple commands that can fit into person's memory. Can anyone enter as a student at the Citadel ? Or must one be recommended by Lords or a powerful institution?   Aemon Targaryen was a son of a King, so the Citadel would let him access without questioning. Samwell Tarly was sent by Lord Commander Jon Snow, from the Night's Watch. , a novice of the Citadel. A common born but it is not known how he entered the Citadel. We can notice that he is not a good student, but has not been fired.    I don't remember how the other Maesters were recruited, if they are always relatives of Lords, Kings... Can any lowborn commoner just "knock the door", and enter the Citadel? I really tried hard to do this on my own, but I keep failing ... Using multi-column and multi-row commands confused me. I know some of you might frown upon such a request but I really need help getting this table implemented .. So I'm sorry if I upset anyone.  My preamble has :   \documentclass[12pt,a4paper]{report} \usepackage[lmargin=3.81cm,tmargin=2.54cm,rmargin=2.54cm,bmargin=2.52cm]{geometry} \linespread{1.5}     Is there a way I can convert this from excel to LaTeX form ? I tried an add-in before but it didn't work for me. Also online generators cannot combine cells the way I want them (unless I didn't find a better table generator ? ) How do I brew an X potion? Proving $\sqrt{ab} = \sqrt a\sqrt b$ I am currently in high school and we are studying radicals. I had asked my math teacher why $\sqrt{ab}=\sqrt{a}\sqrt{b}$ (for all a,b>0) and he tries to prove it by arguing that $a^{1/2}*b^{1/2}=(ab)^{1/2}$ (an exponent law). However, I find this proof problematic since $x^{1/2}$ is simply defined as $\sqrt{x}$, so the reasoning is circular.   My view is that $\sqrt{ab}=\sqrt{a}\sqrt{b}$ because once we square both sides we get $ab=ab$. Since we're obviously referring to the positive root, and the function $f(x) = \sqrt{x}$ is injective, it necessarily follows that the original expressions $\sqrt{ab}$ and $\sqrt{a}\sqrt{b}$ are equivalent because for injective functions it is not possible to map distinct elements in the domain to the same element in the range ($ab$). Hence they are equivalent expressions.   My question therefore is, is my proof valid and/or rigorous (I find it convincing but maybe it's wrong; I just want to be clear) and secondly was my teacher's proof correct? How do I get a Cron like scheduler in Python? I'm looking for a library in Python which will provide at and cron like functionality.  I'd quite like have a pure Python solution, rather than relying on tools installed on the box; this way I run on machines with no cron.  For those unfamiliar with cron: you can schedule tasks based upon an expression like:    0 2 * * 7 /usr/bin/run-backup # run the backups at 0200 on Every Sunday  0 9-17/2 * * 1-5 /usr/bin/purge-temps # run the purge temps command, every 2 hours between 9am and 5pm on Mondays to Fridays.   The cron time expression syntax is less important, but I would like to have something with this sort of flexibility.   If there isn't something that does this for me out-the-box, any suggestions for the building blocks to make something like this would be gratefully received.  Edit I'm not interested in launching processes, just "jobs" also written in Python - python functions. By necessity I think this would be a different thread, but not in a different process.  To this end, I'm looking for the expressivity of the cron time expression, but in Python.   Cron has been around for years, but I'm trying to be as portable as possible. I cannot rely on its presence. Can a single PS Plus subscription be used on two different profiles on the same PS4? Both my girlfriend and I play on the PS4 at home, so there are 2 profiles on PS4. When we set our profile in PS4, we set the console as the primary console.  Now we want to buy PSN Plus subscription, and I want to know if the same subscription can be used in both profiles, or  only in one.   In case the same subscription can be used in both profiles, how can we set it up? Or is automatic? How do I prevent object distortion when applying a curve modifier? How to make all faces flip to the right/consistent direction? What's happening is that I have tried to flip the normals, inside and out, but there are always faces which have to be flipped!  Normals recalculated   Inside View    If I flip them again, all faces get flipped and end up horrible and, not working at all. So how can I make all of the faces flip to the right direction? I run a large bulletin board and I get spammers every now and again. My moderation team does a good job filtering them out but every time I IP ban them they seem to come back (I'm pretty sure it's the same person on some occasions, as the post patterns are exactly the same as are the usernames) but I'm afraid to ban them by IP address every time.  If they are on a dynamic IP address, I could be banning innocent users later down the line when they try to get to my forum through SERPs, but if I ban only via static IPs I know that I'm only banning that one person.  So, is there a way to properly determine if an IP address is static or dynamic?  Thanks. Who is the Community user? Is there a way to convert a List(of Object) to a List(of String) in c# or vb.net without iterating through all the items? (Behind the scenes iteration is fine – I just want concise code)  Update: The best way is probably just to do a new select  myList.Select(function(i) i.ToString()).ToList();   or  myList.Select(i =&gt; i.ToString()).ToList(); How do I install an R package from source? The problem statement is: for $a,b$ coprime, prove that there exists positive integers $x,y$ such that $ax-by=1$.  The question is form Arthur Egnels' problem solving text, and from the chapter on the pigeonhole principle.    My question: His proof begins by generating a list $a,\ldots,a(b-1)$, where he then points out that each element in this list(which proceeds sequentially) doesn't have remainder $0\pmod{b}$.  He then shows that we arrive at a contradiction if we assume that we also don't get remainder $1$ in this list $\bmod b$.  To show the contradiction in this last statement, he first states that we would have positive integers $p,q $ where $ 0&lt;p&lt;q&lt;b$ so that $pa\equiv qa \pmod{b}$.  He then goes on to point out that since we $a$ and $b$ are coprime, we then have that $b| q-p$.  I understand the last part here, but don't see why we get $pa\equiv qa \pmod{b}$.  Thanks in advance