Cyclic vector implies commuting linear operator is a polynomial
Prove that $T$ has a cyclic vector iff its minimal and characteristic polynomials are the same
In the annual rep league, why isn't the difference between rep earned this year and total rep always the same? In the  here on Meta Stack Exchange, it shows that I've earned 9,876 rep this year, and 10,359 rep overall. (It's updated every day at 0:00 UTC, and I so far have earned 1 rep today, since an answer I downvoted a while ago was deleted.)  The difference between my total rep and the rep I earned this year is 483, which means 483 of the rep I've earned is rep I earned in 2017 or earlier. This amount normally never changes, though I can see some reasons why it might change, such as if rep is recalculated and something affects a rep event occurring in a previous year (e.g. a vote from last year was invalidated due to user deletion, a post I made at that time was deleted or undeleted, etc.)  This amount, 483, was a recent change; it used to be 461. Ordinarily, I'd chalk this up to a rep event from last year being recalculated, but I've gone through my full rep history and I can see nothing of the sort happening. On the other hand, on the same day that this number changed from 461 to 483,  was undeleted, recalculating my rep and giving me back 22 (which happens to equal 483-461). It therefore makes sense if that post was from a previous year. However, the post is from this year, so it doesn't make very much sense.  Is there a bug in the way yearly rep in the rep league is calculated, in that rep regained from an undeleted post made the same year is not re-added to the rep earned this year (while it is subtracted properly upon deletion)? Or is there some other event that occurred in the three weeks before 2018 that I had my account that caused this difference to change, that I'm not seeing, and the change being equal to that is just a coincidence? Options for crowdsourcing GIS data? I am looking at collecting spatial information from a range of stakeholders to identify opportunities within a landscape.  However, the data that we are collecting are not presently mapped and so we wish to use an interactive method to get this information onto a map.  Which online websites/methods would people recommend for circulating to a non-technical audience to try and get them to map what they know?    Where the emphasis is on ease of use by the stakeholder. Seeking open raw LiDAR or laser scanner data? Show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$. The problem is:  Suppose that $f(z)$ is continuous on a domain $D$ that contains the unit circle, and that $f(z)$ satisfies: $$\vert f(e^{i\theta})\vert \leq M \; \forall \theta \, \in [0, 2\pi )$$ and $$\bigg \vert \int _{\vert z \vert=1} f(z) \text{d}z \bigg \vert =2\pi M $$ show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$.  I've tried lots of different approaches such as defining a function $g(z)=\frac{f(z)}{\bar{z}}$ and trying to show that $g'(z)=0$ so that $g$ is constant on the circle, or trying to use some bounding theorems for analytic functions to show that $g$ is constant but this requires $g$ to be analytic. Some other methods were using the Cauchy-Riemann equations but I couldn't find any reason why $g$ should be analytic.  Any help would be greatly appreciated, thanks! Let $M_n$ denote the vector space of all $n\times n$ real valued matrices and for any matrix $A\in M_n$ let $A^t$ denote its transpose. Define the linear map $T:M_n\rightarrow M_n$ as $$T(A)=A+A^t.$$ What is the minimal polynomial of $T$? What are the eigenvalues and the corresponding eigenspaces of $T$? What is the characteristic polynomial of $T$?  I learned about minimal polynomials of operators from $\textit{Linear Algebra Done Right}$ by Axler. In his description he explains how in order to find the minimal polynomial you must find the smallest positive integer $m$ such that $T^m$ is a linear combination of $I, T, T^2, \ldots, T^{m-1}$. That is, the smallest integer $m$ such that $$T^m = a_0I+a_1T+a_2T^2+ \cdots +a_{m-1}T^{m-1}.$$  As for the eigenvalues...since we are working with real-valued matrices, then $A$ and $A^t$ have the same eigenvalues, so if $\lambda \in \mathbb{R}$ is an eigenvalue of $A$ (and hence of $A^t$) wouldn't $2\lambda$ be an eigenvalue of $A+A^t$? I apologize if I am saying utter nonsense... How to compute $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!\!\bmod 46,$ for power tower height $2020$? What is the remainder of $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!$ divided by $46$?  The level of powers is $2020$.  First there is no parenthesis so it means 3 power of 3 which is also power 3 and so on 2020 times  Second I think that we can use Fermat's Little Theorem but I don't know how, and maybe there is a better way. What compiler warning level do you recommend for different C/C++ compilers?  gcc and g++ will let you get away with a lot on the default level.  I find the best warning level for me is '-Wall'.  And I always try to remove fix the code for the warnings it generates.  (Even the silly ones about using parenthesis for logical precedence rules or to say I really mean 'if (x = y)')  What are your favorite levels for the different compilers, such as Sun CC, aCC (HPUX ?), Visual Studio, intel?  Edit:  I just wanted to point out that I don't use "-Werror" (but I do understand it's utility) on gcc/g++ because, I use:   #warning "this is a note to myself"   in a few places in my code.  Do all the compilers understand the #warning macro? Why there is a delay between when a key is pressed and when it is accepted? Java in Eclipse: Where do I put files on the filesystem that I want to load using getResource? (e.g. images for an ImageIcon) I know the file needs to be where the getClass().getResource(filename) can find it, but I don't know where that is.  I'm interested both in where to put the files on the filesystem itself, and how to go about using Eclipse's functionality to set up the resources. Symbolic Link: Creating link from "NTFS Partiton" to ~/Desktop Finding basis of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ over $\mathbb{Q}$ Should methods in a Java interface be declared with or without a public access modifier? Should methods in a Java interface be declared with or without the public access modifier?  Technically it doesn't matter, of course. A class method that implements an interface is always public. But what is a better convention?  Java itself is not consistent in this. See for instance Collection vs. Comparable, or Future vs. ScriptEngine. changing notification emails from WordPress @mydomain.net to something else How do I change notification emails address from WordPress @mydomain.net to something else.  I want to do this because WordPress @mydomain.net ends up getting flagged as junk mail.  Thanks  Daniel Determine if Type is a pointer in a template function Simple request, really.  Can we add a detection for the     Please avoid extended discussions in   comments. Would you like to   automatically move this discussion to   chat?   prompt, so that if one of the users in the "extended discussion" has under 20 rep it doesn't appear?  I think a lot of the users this apply to will have low rep and it may take some back and forth to get good info out of them.  New suggestion: how about if a sufficiently high-rep user is clicking on the link, the low-rep user gets a field promotion sufficient to chat? Low-rep users are particularly prone to post questions to lead in this direction of needing to move to chat. What is the maximum size of a web browser's cookie's key? What is the maximum size of a web browser's cookie's key?  I know the maximum size of a cookie is 4KB, but does the key have a limitation as well? Powering down the display in XP with a command Is there a command to power down the display in Windows XP? That is, a command that causes the video card to stop sending a signal to the monitor until input is received, much as power management does after a time. Add sort by relevance to search jobs In Careers 2.0, individuals can choose "search jobs" and sort by date posted, job title, and company name.  However, Careers 2.0 already knows what tags I'm proficient in, and knows the tags associated with the job postings.  A sort option, relevance, should be added so that jobs matching my skill set, as demonstrated on stack exchange, will float to the top. to Global Variables inside functions Will I get OTA updates after installing Android L Developer preview?