Let $A,B \in M(n,\mathbb C)$ ; if $\exists P \in M(n, \mathbb C)$ and $f(X) , g(X) \in \mathbb C[X]$ such that $A=f(P) , B=g(P)$ then $AB=BA$ . Now suppose $AB=BA$ ; then does there exist $P \in M(n,\mathbb C)$ and $f(X) , g(X) \in \mathbb C[X]$ such that $A=f(P)$ and $ B=g(P)$ ? If this is not true in general , then what if we assume that both $A,B$ are diagonalizable ? Is it true then ?
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.
How do I install Skype? I have an iPhone 6 (with proper iTunes backup) where all of my contacts where, but I bought a new iPhone 7 and had to initialize it as new iPhone, due to several time-related constraints, so now I have a few but new contacts on my new phone, but none of the older ones, but would be the best practice oriented, for just recovering the old contacts from the iTunes backups?  PS: I still have the old iPhone 6, so it could be used as "temporary storage" if needed. Find the greatest value of $a^2\cdot b^3$ where $a+b=10$ Find the greatest value of $a^2\cdot b^3$ where $a,b$ are positive real numbers satisfying $a+b=10$. Determine the values of $a,b$ for which the greatest value is attained.  At first I tried to find the value by putting $a=10-b$, I found a expression, but I got not way to do anything.   If we apply A.M-G.M. inequality, I think we found the minimum value or something else. It is a pre-Olympiad level problem. So I need some answers or some good hints. Evaluate $\prod \frac {2k - 1} {2k}$ I came across to this: evaluate $$\prod_{k = 1}^{n} \frac {2k - 1} {2k}.$$  I brought it to a closed-form expression in asymptotics but I suspect my asymptotics is bad and I also want to see different solutions, hence the question.  My original solution: Write   $$\begin {eqnarray} \prod_{k = 1}^{n} \frac {2k - 1} {2k} &amp; = &amp; \exp \left (\sum_{k = 1}^{n} \log \left (1 - \frac {1} {2k}\right) \right) \nonumber \\ &amp; = &amp; \exp \left (- \sum_{k = 1}^{n} \sum_{m = 1}^{\infty} \frac {1} {(2k)^m m}\right). \end {eqnarray}$$  Let $$S = \sum_{m = 1}^{\infty} \frac {1} {(2k)^m m} \tag {1}.$$ Multiply both sides of $(1)$ by $2k$ to get $2k S &lt; S + 1$, and by $4k$ to get $4kS &gt; S + 2$. Hence, $$\frac {2} {4k - 1} &lt; S &lt; \frac {1} {2k - 1}.$$ Then, $$\frac {1} {2} \left( \log (4n - 1) - \log 3 \right) - 2 + O \left(\frac {1} {n^2}\right) &lt; \sum_{k = 1}^{n} \sum_{m = 1}^{\infty} \frac {1} {(2k)^m m} &lt; \frac {1} {2} \log (2n - 1) - 1 + O \left(\frac {1} {n^2}\right).$$ Hence, $$\sqrt {\frac {c} {2n - 1}} &lt; \prod_{k = 1}^{n} \frac {2k - 1} {2k} &lt; \sqrt {\frac {3 c^2} {4n - 1}},$$ where $c$ is a constant. Any better approach? How long does it take to unlock another frame by playing one of the base frames? I'd like to get an advanced frame in Firefall. I've been told that I can use "Pilot Tokens" to unlock other frames without having to pay money.  How soon would I be able to unlock an advanced frame? Do I have to level a base frame all the way to level 40? More than once? Charge controller for Li-ion or Li-Polymer battery I am trying to work out the good charger for Mobile phones, which can serve as good performer for various smart phones out there in market these days &amp; mostly using Li-ion battery. Charging has 4 stages -- "the initial trickle-charging, then constant current charging, then constant voltage charging &amp; finaly Top-up charge" -- all this is controlled &amp; monitored by a control circuit/IC of charger. So, my concern is if such circuit/module/IC is already there in Mobile phones &amp; taking care of all protections to battery/phone including overheating during charge OR The charger designer have to incorporate these circuits in Charger itself? If former case, that means we just need to make a charging source giving ample current &amp; 4.75V-5.25V voltage, like a high-powered USB output port would gives. My feeling is such stages-wise charging &amp; control/monitor circuitry is already within the phone set, i just want to reconfirm. Confidence interval interpretation in linear regression when errors are not normally distributed Travel grant for undergraduate researcher for attending conference I am an undergraduate student doing independent research. My research work is selected for presentation at a very reputed conference in the US. I am from India. I cannot afford to attend the conference at all. How should I fund my trip? The travel grant for students available from the conference is very low. It can hardly buy me a one way ticket. What are my options? Simple UV Mesh Unwrapping Scale issue What is a word for getting an award in a competition or being one of the top three? Is it possible to set the "urgent" flag in Gmail for outgoing mail? I'd like to be able to flag time-sensitive emails as such. What are the obj and bin folders (created by Visual Studio) used for? Calculating p-values for two tail test for population variance A sporting goods manufacturer claims that the variance of string tensions for any decent tennis racquet should be about 9 pounds. The string tensions of 18 randomly selected tennis racquets produced a variance of 8.13 pounds. Find the p-value to test the manufacturer's claim (assume the population is normally distributed).  I tried attempting the question as follows: ${H_o}: {\sigma ^2} = 9$ ${H_1}: {\sigma ^2} \ne 9$ We are also given the following: $n=18$ $s^2 = 8.13$ Test Statistics: $\chi^2 = {{(n-1)S^2}\over{\sigma^2}}$ Calculating Test Statistics under $H_o$ gives us: $\chi^2 = {{(18-1)8.13}\over 9} = 15.35667$  For calculating the p-value I proceeded as follows: $p-value = 2\min\{P[\chi^2&gt;15.35667],\ P[\chi^2&lt;15.35667]\}\ =2\times 0.430192 = 0.860384$  I am not certain if my method for calculating the p-value is correct or not. Can someone please tell if the above process is valid?  Thanks in advance! Is causality a formalised concept in physics? Should I use  tag for icons instead of ? Setting or modifying a (system wide) environment variable in cmd.exe I am looking for a convenient way to add and/or modify and/or delete an environment variable from the command line. Particularly, I find myself at times in situations when I have to add a few variables in cmd.exe.  I'd be grateful if someone showed me a non-GUI way to modify (that is: to add a new directory to) the %PATH% variable.  The change should be be permanent, not just for duration of the cmd session. Recently I decided to use only Ubuntu and there is no Windows in my laptop. When I turn on my laptop the GRUB menu is showing but I don't need to choose OS anymore.   How can I remove or hide this? And after removing this how could I access to recovery mode? Print escaped representation of a str "Baggage" versus "luggage" I have the feeling that luggage is more closely associated with vacation travel, whereas baggage is for general transportation.  Or... are they just exact synonyms? Stuff like polls, recommendations based on subjective constraints, puzzles, webcomics etc. do not belong on the serious main SE sites, where professionals should be considered at work and having just a few spare minutes ("code's compiling") to answer questions, so they should not be distracted by such things. However, I'd also like to have a home for these things still using the SE engine. For the reason I mentioned before, this needs to be a separate place though. Let's call it four.[sitename-here].com.