I am confused as to why eigenvalues are the principal components. What is the intuition behind finding the eigenvectors of the covariance matrix for PCA?
In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues.   I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like a machine. But I didn't understand it. I didn't get the purpose of it. I didn't get the feel of it.        I strongly believe in the following quote:     You do not really understand something unless you can explain it to your grandmother. -- Albert Einstein   Well, I can't explain these concepts to a layman or grandma.   Why PCA, eigenvectors &amp; eigenvalues? What was the need for these concepts? How would you explain these to a layman?
On the nilpotence of the matrix $AB-BA$ I am curious about the proof of these two integrations. It would be great if you could point me in the right direction : $ \int{e^{ax}\cos({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\cos(bx)+b\sin(bx)] + C $ &amp; $ \int{e^{ax}\sin({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\sin(bx)-b\cos(bx)] + C $ Solve $x=y\frac{dy}{dx}-\left(\frac{dy}{dx}\right)^{2}$ My question is whether $x/x$ is always equal to $1$. I am mostly intersted in real numbers and particularly wonder whether $x/x$ is defined at $x=0$.  On one hand the division should simplify to $1$, on the other hand you should not be allowed to divide by zero.   I have been trying to find whether the simplification 'goes first' or whether the division causes trouble first, but it has proven impossible for me to find usefull search terms.     Note that this question arose after reading  and its first comment. So basically, I'd like to use iwconfig to connect to my wifi network when I'm not inside X. But I just don't see a way to do it in the man page. Any idea? Headline Language Is there a particular term for the abbreviated language used in headlines (the removal of at least articles and conjunctions)? Missing argument 2 for wpdb::prepare() Issue I'm having to update a WP installation for a community organisations from version 2.9.1 to the current 3.9.1.   The installation has a couple of custom built plugins that aren't documented.  I've just performed the WP version update and now I see a warning (Missing argument 2 for wpdb::prepare()) on every page that relates to the following line in the code of one of these custom built plugin:  $rows = $wpdb-&gt;get_results($wpdb-&gt;prepare("SELECT id, name FROM ".$table_name));   From what I've read online, this is due to a change in the coding, that requires 2 values to be sent to wpdb::prepare(), but I've no idea how the line of code given above, needs to be changed.  If someone could help me out, I'd really appreciate it. QT MySql connectivity using Windows XP, Qt Creator 4.5.2(windows 32 bit) How do I create a reflective white material? I'm pretty new to this and this is my first question here, any help would be greatly appreciated.  So basically what I want to achieve is a floor material as seen in this photo...    ...the reflections are very crisp but dim.  With a glossy shader I get the following results:    First picture: reflections are blurry  Second picture: roughness of the material looks right but reflection is bright red  Third picture: If tinted in any other color than white, it seems to produce the result I want. Reflection is sharp and gets tinted blue. What I need is the reflection to be tinted white.  Thanks for any help. How do you define an arbitrary sort sequence for views? Let's assume a very simplified situation. There is only one seller asking for 10$ and there is only one buyer bidding 20$. At what price would the transaction go?   Some answers say it won't, because the ask price has to match the bid price. () Other answers say, that the time ordering is important. () Which answer is correct? What is the final price in the presented situation? Let $a_n$ be a sequence s.t $a_1 > 0 \land a_{n+1} = a_n + \dfrac{1}{a_n}$. Prove that $a_n$ is increasing and tends to infinity Let $a_n$ be a sequence s.t $a_1 &gt; 0 \land a_{n+1} = a_n + \frac{1}{a_n}$. Prove that $a_n$ is increasing and tends to infinity.  Proof:   Consider $a_{n+1} - a_n$:  $a_{n+1} - a_n = a_n + \frac{1}{a_n} - a_n = \frac{1}{a_n}$ This is greater than $0$. Thus, $a_n$ is increasing.  Now this is where I need some help. I would like to say that $a_n$ is unbounded and then conclude that monotone and unbounded implies tending to infinity.  Maybe by contradiction? What techniques have been used to repeat geometry or other detail to the infinite horizon? It is often stated, particularly in popular physics articles and videos, that if one measures a particle A that is entangled with some other particle B, then this measurement will immediately affect the state of the entangled partner.  For example, if Alice and Bob share an entangled pair of electrons and Alice measures her spin in the $x$ direction, then Bob's spin will also end up spinning in that direction, and similarly if she measures in the $z$ direction. Moreover, the effect will be instantaneous, regardless of the spatial distance between the two particles, which seems at odds with special relativity.  Can I use a scheme like this to communicate faster than light? consider positive numbers $a_1,a_2,a_3,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$. does the following in-equality holds and if it does then how to prove it  $\left[(a_1+b_1)(a_2+b_2)\cdots(a_n+b_n)\right]^{1/n}\ge \left(a_1a_2\cdots a_n\right)^{1/n}+\left(b_1b_2\cdots b_n\right)^{1/n}$ What is the use of the yield keyword in Python? What does it do? For example, I'm trying to understand this code1: def _get_child_candidates(self, distance, min_dist, max_dist):     if self._leftchild and distance - max_dist &lt; self._median:         yield self._leftchild     if self._rightchild and distance + max_dist &gt;= self._median:         yield self._rightchild    And this is the caller: result, candidates = [], [self] while candidates:     node = candidates.pop()     distance = node._get_dist(obj)     if distance &lt;= max_dist and distance &gt;= min_dist:         result.extend(node._values)     candidates.extend(node._get_child_candidates(distance, min_dist, max_dist)) return result  What happens when the method _get_child_candidates is called? Is a list returned? A single element? Is it called again? When will subsequent calls stop?   1. This piece of code was written by Jochen Schulz (jrschulz), who made a great Python library for metric spaces. This is the link to the complete source: [Module mspace][1]. So I have some ISO stock options (yeah) and three kids in college (boo) so I'm trying figure out the best tax strategy. I was hoping someone help me through the following scenario  Strike Price: $10, 2014 exercise price: $20, 2015 price: $25, 2016 price $30   Lets' assume that all of this happens 2+ years after the grant date and that my "normal" tax bill is right on the brink of the AMT threshold, that the AMT tax rate is 28% and the long term capital gains tax is 15%.   Now I want to run the following scenario  2014: exercise 1000 options 2015: exercise 1000 options, sell the previous years' option after "one year and one day" 2016: sell the 2015 options after "one year and a day"    Tax wise that would look like this  2014: AMT on $10000 of bargain element so $2800 AMT 2015: AMT on $15000 bargain element $4200, long term gain tax on $15000 = $2250 2016: long term gain tax on $20000 = $3000    What's the overall tax liability in each year? I can't figure out how the AMT carries forward when the options are finally sold.   Any good pointers to modeling software, spreadsheets or decent papers would also be appreciated. "May I go to the bathroom?" and "I asked if I might go to the bathroom."   The modal auxiliary 'may/might' is used to ask permission.     "He may not have understood your question." and "He might not have understood your question."   The same modal auxiliary is used for uncertainty.  Why? Dynamically access object property using variable I have several files that I ftp each hour.  The receiving system needs to have some sort of identifier that they came from the same batch, so I would like to append a timestamp to the filename.  That alone is fairly easy, but since I want each file to have the same timestamp (so it can serve as the batch identifier), I cannot figure out how to do this.  So, I have:  file1.txt file2.txt file3.txt   And I want to have:  file1_20141110184303.txt file2_20141110184303.txt file3_20141110184303.txt   Any use of the date +%Y%m%d%H%M%S after the first use will obviously result in a different values in seconds, so I would like to have the first timestamp appended to the remaining files.