When working with a general matrix $A=$ \begin{bmatrix}a &amp; b\\c &amp; d\end{bmatrix},  I find that the eigenvalues are:  $\lambda = \frac{d+a \pm \sqrt{(d-a)^2+4bc}}{2}$  I then find that eigenvectors, $e=$ \begin{bmatrix}-b\\a-\lambda\end{bmatrix}.  I also know that under iteration with initial vector $x_n$ = \begin{bmatrix}x_0\\y_0\end{bmatrix},  we have $x_{n+1} = Ax_n$.  The matrix $A$ converges to a line with a slope $m$.   I am told that the slope of this line is in the same direction as the eigenvector associated with the largest eigenvalue.   How do I prove this?
Suppose I am given a $2$x$2$ matrix $A=$ \begin{pmatrix} a &amp; b \\ c &amp; d \end{pmatrix}  And an initial vector $x_n$ = \begin{pmatrix} x_0 \\ y_0\end{pmatrix}.  Under repeated iteration $x_{n+1} = Ax_n$, we find that $x_{n+1}$ converges to a line in the direction of the $eigenvector$  of $A$ of the form $X$= $[u,mu]$ where $m$ is the slope of the line.  I know that this process somehow also involves the largest $eigenvalue, \lambda,$ of $A$, though I am unsure how it connects.  My question is: How does the largest eigenvalue connect to this repeated iteration, how do I find the slope of the line of convergence $m$, and what is meant by the "direction of the eigenvector"?
"Over vs Above" and "Under vs below" Is is possible to say that "Above" means "over" and "Under" means"below", but not the opposite?  According to my grammar:  Over - In a higher position (close or touching) Above - In a higher position  Under - In a lower position (close or touching) Below - In a lower position  Example:     There's a helicopter flying round above / over the school.   I would interpret this sentence as "above" meaning almost touching the school or not. And If I hear "over", the helicopter is almost touching school.  This makes me think about "over" and "on". "On" means that something is touching something, but if "over" can mean also "touch".Well, I would never use "over" to mean "touching something", I would use "on" instead. I've been asked to show that:  $n(n+1)(2n+1) \equiv 0 \pmod 6$  I found in a previous question that:  $n(n+1)$ was divisible by $2$ and resulted in an even number e.g  $n(n+1) \equiv 0 \pmod 2$  so I figured I needed to find:  $(2n+1) \equiv 0 \pmod 3$ in order to complete  $n(n+1)(2n+1) \equiv 0 \pmod 6$  but I am unsure on how to find $(2n+1) \equiv 0 \pmod 3$  Is this the right way to find the mod 6 and if so could you tell me how I could find  $(2n+1) \equiv 0 \pmod 3?$ Where do object file "Version References" come from? Get files in a directory sorted by last modified? python: restarting a loop What determines how much experience I get from a hearthstone match in play mode? Sometimes I win handily and get a little experience, sometimes I squeak out a victory and get a lot.  How is experience earned? Best word for "unable to change" How to approach a student with bad hygiene? I teach engineering at a community college in the US. I currently have a student with very poor hygiene. He stinks pretty badly, has obviously greasy hair, doesn’t change his clothes frequently, and I usually seat him in the back of the room (during exams and labs, when I have power over seating arrangements) so I don’t have to smell his (relatively strong) body odor.  This student has also spoken with a colleague of mine about his low self-esteem. He’s worried that he isn’t making friends, and doesn’t know if the people he hangs out with are hanging out with him because they want something from him (help with classwork) or because they are truly friends.   In my opinion, this student follows me around like a puppy, and he tries too hard to impress people (students and faculty alike). I am a young-ish female faculty member and I try to create very large boundaries, especially with my male students. As an example, he frequently finds the need to explain to me (in way too much detail) why he gets questions wrong on exams (I don’t care, and I don’t need to know, which I’ve told him several times with no success).  So, I would like to mention to this student that his hygiene is off-putting and isn’t helping him out in the friends department. However, because he already follows me around like a puppy dog, I don’t want to encourage any more interaction from him. Additionally, it frankly embarrasses me to have to bring up hygiene issues with somebody in their late teens / early twenties. How can I bring this issue up while still maintaining my boundary as a female faculty member who doesn’t want her male students following her around and asking her for advice all the time?  I was hoping that I would be finished up with this student by now, but I am the only professor who teaches Electrical Engineering courses, so I can’t recommend that he take classes with someone else, or ask another colleague to discuss this with him. He will be taking another of my courses next semester and I don’t personally know any of his other intended professors for next semester to ask them to bring it up. Advantages of ROC curves What is the advantages of the ROC curves?  For example I am classifying some images which is a binary classification problem. I extracted about 500 features and applied a features selection algorithm to select a set of features then I applied SVM for classification. In this case how can I get a ROC curve? Should I change the threshold values of my feature selection algorithm and get sensitivty and specificity of the output to draw a ROC curve?  In my case what is the purpose of creating a ROC curve? Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ Gravitational slingshot of light using a black hole/massive object Wikipedia has this page on  using planets. In some cases this effect was used to accelerate the spacecraft to a higher velocity. This diagram shows this in a very oversimplified manner.   That got me thinking that if light is affected by gravity, and if it slingshots around the black hole/massive object, can't it gain a higher speed than $c$? What limitations are stopping it from doing this?  Forgive me if the answer to this question is pretty straight-forward or staring-you-in-the-face kind. I haven't fully understood the mechanics of the gravitational slingshot fully yet, but I couldn't wait to ask this. Note: If it is possible (I highly doubt that!), could you provide an explanation using Newtonian mechanics, I'm not very familiar with general relativity because I'm in high school. Should I say "She is in the park" or "She is at the park"? I am really confused.  Which preposition is correct?     She is in/at the park.   They are in/at the park.   I am in/at the park.   Should I use in or at in these sentences? Search and download audio-only from youtube Is there a way to search and then download from YouTube, with the situation that I only need to download the audio only, not the video. Need help recalling name of sci-if story about earth evolving into electronic life forms Read a story once about earth being abandoned for long periods of time. Upon the return of man to earth all life forms, flora and fauna, have evolved into intricate electronic forms of themselves.   Would like to find that story and it's name. Identifying records in MC to delete I'm trying to delete a large number of records from SFMC to avoid being charged extra. I've run several different SQLs to help identify the records I don't need, but when I run the Contact Count report, it still shows approx 100K billable contacts and approx 60K distinct email addresses.  Here are the SQLs I've tried:   No Opens in past 5 days Leads/Contacts not in SFDC  SubscriberKey as Email Address Leads/Contacts where SubscriberKey is Empty Unspecified records (DE populated by MC Support)    These efforts reduced the count a good amount, but not as much as I was expecting.   Has anyone else gone through this process before and if so, what queries can you share that might help my case? Array only adds duplicate values Cursor hitting "border" between differently sized monitors I'm not sure if I'll be to explain this right. I have a dual monitor display on Windows 7 and my left monitor (laptop) has a lower vertical resolution than the right monitor (regular LCD monitor). So obviously, if I stick these monitors together, there's a part in the upper-left corner of my LCD monitor where cursor cannot go to the other monitor. I remember there was a trick to make this gap disappear, and make it seem like the monitors are of equal height.  Any suggestions?  By the way, I am also using UltraMon. How do you set a default value for a MySQL Datetime column?  In SQL Server it's getdate(). What is the equivalant for MySQL? I'm using MySQL 5.x if that is a factor. Introduction to AdS/CFT Why do eukaryotic organisms have introns in their DNA? We touched on introns and exons in my bio class, but unfortunately we didn't really talk about why Eukaryotes have introns. It would seem they would have to have some purpose since prokaryotes do not have them and they evolved first chronologically, but I could easily be wrong. Did the junk sections of DNA just evolve there by some sort of randomness or necessity as opposed to an actual evolutionary advantage? 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