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No Wifi, HP Pavillion 15 Ubuntu 14.04 Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$? Consider square matrices over a field $K$. I don't think additional assumptions about $K$ like algebraically closed or characteristic $0$ are pertinent, but feel free to make them for comfort. For any such matrix $A$, the set $K[A]$ of polynomials in $A$ is a commutative subalgebra of $M_n(K)$; the question is whether for any pair of commuting matrices $X,Y$ at least one such commutative subalgebra can be found that contains both $X$ and $Y$.  I was asking myself this in connection with frequently recurring requests to completely characterise commuting pairs of matrices, like . While providing a useful characterisation seems impossible, a positive anwer to the current question would at least provide some answer.  Note that in many rather likely situations one can in fact take $A$ to be one of the matrices $X,Y$, for instance when one of the matrices , or more generally if its  (so coincides with the characteristic polynomial). However this is not always possible, as can be easily seen for instance for diagonal matrices $X=\operatorname{diag}(0,0,1)$ and $Y=\operatorname{diag}(0,1,1)$. However in that case both will be polynomials in $A=\operatorname{diag}(x,y,z)$ for any distinct values $x,y,z$ (then $K[A]$ consists of all diagonal matrices); although in the example in  the matrices are not both diagonalisable, an appropriate $A$ can be found there as well.   I thought for some time that any maximal commutative subalgebra of $M_n(K)$ was of the form $K[A]$ (which would imply a positive answer) for some $A$ with minimal polynomial of degree$~n$, and that a positive answer to my question was in fact instrumental in proving this. However I was wrong on both counts: there exist (for $n\geq 4$) commutative subalgebras of dimension${}&gt;n$ (whereas $\dim_KK[A]\leq n$ for all $A\in M_n(K)$) as shown in , and I was forced to correct  in the light of this; however it seems (at least in the cases I looked at) that many (all?) pairs of matrices $X,Y$ in such a subalgebra still admit a matrix $A$ (which in general is not in the subalgebra) such that $X,Y\in K[A]$. This indicates that a positive answer to my question would not contradict the existence of such large commutative subalgebras: it would just mean that to obtain a maximal dimensional subalgebra containing $X,Y$ one should in general avoid throwing in an $A$ with $X,Y\in K[A]$. I do think these large subalgebras easily show that my question but for three commuting matrices has a negative answer.  Finally I note that  to the cited MO question mentions a result by Gerstenhaber that the dimension of the subalgebra generated two commuting matrices in $M_n(K)$ cannot exceed$~n$. This unfortunately does not settle my question (if $X,Y$ would generate a subalgebra of dimension${}&gt;n$, it would have proved a negative answer); it just might be that the mentioned result is true because of the existence of $A$ (I don't have access to a proof right now, but given the formulation it seems unlikely that it was done this way).  OK, I've tried to build up the suspense. Honesty demands that I say that I do know the answer to my question, since a colleague of mine provided a convincing one. I will however not give this answer right away, but post it once there has been some time for gathering answers here; who knows somebody will prove a different answer than the one I have (heaven forbid), or at least give the same answer with a different justification. I'm trying to wrap my head around the math/physics of how illuminance on the film/sensor plane remains equal for a given f-number regardless of the focal length of the lens.  Since a longer lens will have a larger entrance pupil at a given f-stop than a shorter lens at the same f-stop, how can the light falling on the film/sensor be equivalent?  Empirically, I know this to be true.  Say I take an incident meter reading (not reflected/TTL) off my subject, and get f/5.6 at 1/100s, I'll get proper exposure at the point at which i took the meter reading with that setting whether I use a 50mm lens or a 200mm lens (obviously composition will be different).    Wikipedia's  reads:     A 100 mm focal length f/4 lens has an entrance pupil diameter of 25   mm. A 200 mm focal length f/4 lens has an entrance pupil diameter of   50 mm. The 200 mm lens's entrance pupil has four times the area of the   100 mm lens's entrance pupil, and thus collects four times as much   light from each object in the lens's field of view. But compared to   the 100 mm lens, the 200 mm lens projects an image of each object   twice as high and twice as wide, covering four times the area, and so   both lenses produce the same illuminance at the focal plane when   imaging a scene of a given luminance.   I'm not sure I agree with the explanation on wikipedia.  The film/sensor doesn't care how big the image is that falls outside the bounds of the sensor; the film/sensor "sees" what it sees, and that's it. How can I bump other users' unanswered questions? I'm looking around for answers to my questions on several Stack Exchange sites and the good boy that I am I don't want to ask duplicate questions.  Often I find that there are other people who asked the same questions, sometimes years earlier, but still don't have an answer. I'd figure I could bump these back to top, but if I don't have enough rep for a edit (or comment), how would I do that?  Or, in that case, I should just post a 'newer' version of the same question? How to get exact camera view? So I'm going to be starting a Job here in a few weeks, I've been doing QA for about 6 years now. Pretty much your standard Test Plans/Testing with a little normal SysAdmin stuff that sometimes comes with the task. I've had some good ideas implemented in companies, as far as tracking requirements/test plans/issues differently, and they seem successful. However, this new position will be me laying down the foundation for QA. Right now things are done in what seems to be as needed as far as QA goes, requirements are gathered and sometimes kept in different locations, same with test plans (which are done by developers right now). So my job will be sort of centralizing this, or developing a Flexible way of QA'ing the Software (Web/Mobile development). So it seems the challenge will be centralizing everything, while creating test plans/ or ways of testing similar requirements that is flexible enough to span projects. I've had a few ideas, however, all my previous test work was windows based (So test plans in TFS or Microsoft Test Manager). I wanted to get some ideas from everyone here, or anyone that has started a QA department from scratch. I was also going to see if there were any books on the subject. My thoughts were mainly keeping projects separated for issue tracking (not sure of the software we will use yet). And keeping test plans for common I guess &quot;actions/elements&quot; that are repeated between different customers and executing those. Eventually maybe using selenium to save those &quot;common&quot; elements as test scripts. But I've never set something up like this before solo, so I've been doing quite a bit of reading. Any books/suggestions would be appreciated! General formula for evaluating integrals of the form $\displaystyle\int_0^\infty x^ae^{-bx}dx$ Embedding a link (or other html) in a JSF message launch an exe/process with stdin stdout and stderr? How does Linux kernel compare to microkernel architectures? I read once that one advantage of a microkernel architecture is that you can stop/start essential services like networking and filesystems, without needing to restart the whole system. But considering that Linux kernel nowadays (was it always the case?) offers the option to use modules to achieve the same effect, what are the (remaining) advantages of a microkernel? Please help me to stop auto login in Skype. There is an account that is always login again and again, although I chose to logout and sign in with another Skype name. I tried on Settings/applications/Skype to kill the account and to quit synchronise, but still whenever I open Skype again, I'm logging in automatically. Do I need the original discs to install WoW on a new computer? My boyfriend bought the World of Warcraft Battle Chest, Wrath of the Lich King, &amp; Cataclysm. He downloaded them to our previous computer, but now we have a new one and when he took out the first disk from the battle chest he severely scratched it. Anyway, I tried putting in disk #2, and it said I couldn't do it without disk 1 (which is scratched!).  I've recently put in Wrath of the Lich King (it came with 2 CDs), and it said that I haven't downloaded the original and asked me if I wanted to downloaded. I clicked "yes" -- does that mean its downloading WoW &amp; The Burning Crusade? Or do we need to buy it all again?  We just bought the whole set on June of this year... When is the new ball due for all forms of cricket? Is there a standard time fixed in Test, ODI and T20 formats of Cricket, when a fielding side captain can ask for a new ball?  If the captain of the fielding side doesn't ask for a new ball, will he be forced to have it mandatorily after a certain point by the match umpires? Synchronize Clocks I need to develop a system where there are 4 sensor boxes powered by USB and they have their internal clock. Each of these boxes are connected to a Linux pc and all these PCs are linked with WiFi. API of the sensor box allows me to read their internal precision clock (uSec accuracy).   I like to synchronize all four clocks of the sensor boxes as close as possible. Hopefully within a few micro seconds of each other and compensate for drift over time. The idea is to use the sensor data that is collected at the same time (The data I receive is time stamped with the boxes' internal clock). The latency on usb bus, WiFi, pc timing are all indeterministic.  Is this possible given my setup? How can I see the source of built-in JavaScript functions? Skip certain tables with mysqldump If $p,q$ are positive quantities and $0 \leq m\leq 1$ then Prove that $$(p+q)^m \leq p^m+q^m$$    Trial: For $m=0$, $(p+q)^0=1 &lt; 2= p^0+q^0$  and for $m=1$, $(p+q)^1=p+q =p^1+q^1$.   So, For $m=0,1$ the inequality is true.How I show that the inequality is also true for $0 &lt; m &lt; 1$.   Please help. Print leading zeros with C++ output operator? Mark my own question as favorite? Your content must have a ListView whose id attribute is 'android.R.id.list'