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This question is about an efficient way to compute principal components.   Many texts on linear PCA advocate using singular-value decomposition of the casewise data. That is, if we have data $\bf X$ and want to replace the variables (its columns) by principal components, we do SVD: $\bf X=USV'$, singular values (sq. roots of the eigenvalues) occupying the main diagonal of $\bf S$, right eigenvectors $\bf V$ are the orthogonal rotation matrix of axes-variables into axes-components, left eigenvectors $\bf U$ are like $\bf V$, only for cases. We can then compute component values as $ \bf C=XV=US$. Another way to do PCA of variables is via decomposition of $\bf R=X'X$ square matrix (i.e. $\bf R$  correlations or covariances etc., between the variables ). The decomposition may be eigen-decomposition or singular-value decomposition: with square symmetric positive semidefinite matrix, they will give the same result $\bf R=VLV'$ with eigenvalues as the diagonal of $\bf L$, and $\bf V$ as described earlier. Component values will be $\bf C=XV$.   Now, my question: if data $\bf X$ is a big matrix, and number of cases is (which is often a case) much greater than the number of variables, then way (1) is expected to be much slower than way (2), because way (1) applies a quite expensive algorithm (such as SVD) to a big matrix; it computes and stores huge matrix $\bf U$ which we really doesn't need in our case (the PCA of variables). If so, then why so many texbooks seem to advocate or just mention only way (1)? Maybe it is efficient and I'm missing something? For example, in a novel, if someone is so powerful that no one can stop him, something like this would be said:     He [insert word] in the continent [or some other location].   "Moved unhinderedly" somewhat fits it, but it doesn't really seem like the right term.  The word I'm looking for describes a few things:   Should have an implication that the person himself is powerful That person can go wherever he wishes to, and cannot be stopped because of his strength That person can do whatever he wishes to, and like above, cannot be stopped because of his strength   The other word I have is "swept", i.e.:     He swept through the continent.   I used "swept" because when you sweep the floor, nothing stops you and everything is swept away, similar to the example. However, the dictionary definitions don't match up, so that isn't the word either.  Any suggestions? Use of "would" for subjunctive phrases This has been bugging me for some time; I tried to look for previous questions here but my language tools may not be sharp enough to phrase my query correctly so please forgive me if this has already been posted here.  Speakers of English as a foreign language tend to incorrectly use the word "would" when using some particular subjunctive constructs. Here is  (this time presumably by a native English speaker):     If that would happen, you won’t be alone.     (One correct version being "If that happens, you won't be alone".) I have detected this in speakers of a number of languages so I feel this is more likely to be a feature of the English language itself than a carry-over from their native language.  I have a faint intuition that this is because , and it certainly doesn't help that , but I can neither make that precise nor even really phrase a concrete question. Can someone shed some light on this?    Edit: just to provide some real-world examples of what I mean, here is one example taken from :     Publish or Perish is designed to empower individual academics to present their case for research impact to its best advantage. We would be concerned if it would be used for academic staff evaluation purposes in a mechanistic way.   This is obviously ungrammatical (to me), but I'm not sure how to characterize the mistake, or what the likely causes are (e.g. whether it follows from analogous constructions in other languages). How do you get a bad transcript past Ph.D. admissions? Variable importance from GLMNET Point reflection over a line I'm having trouble understanding the solution presented below (It's from a textbook). I tried to get something similar, but to no avail. Help me find the way he derived those a,b,x2 and y2 expressions.   Point p is a point which we want to reflect (It's a simple structure with two integers x and y representing coordinates on a 2d plane); (x0,y0) and (x1,y1) are coordinates of endpoints of a line.  Expected result is a new point with reflected coordinates.   Point mirror(Point p, int x0, int y0, int x1, int y1) {     double dx,dy,a,b;    long x2,y2;    Point p1; //reflected point to be returned      dx  = (double) (x1 - x0);    dy  = (double (y1 - y0);     a   = (dx * dx - dy * dy) / (dx * dx + dy*dy);    b   = 2 * dx * dy / (dx*dx + dy*dy);     x2  = Math.round(a * (p.x - x0) + b*(p.y - y0) + x0);     y2  = Math.round(b * (p.x - x0) - a*(p.y - y0) + y0);     p1 = Point((int)x2,(int)y2);      return p1;  } As I have seen in other posts, i.stack.imgur.com is blocked by numerous companies' firewalls including my own. I have started the process to get the restriction lifted but to increase the likelihood of it being granted, have only requested that i.stack.imgur.com be unblocked.  Staying within the context of SO, imgur and the hosting arrangement SO has with them, are there any other sub-domains/domains that I should be requesting be unblocked? i.e. p.stack.imgur.com or *.stack.imgur.com?  I'm not sure if images can still be embedded/hotlinked in posts from other image hosts or not but if they can I'm not interested in unblocking those hosts, just the addresses used by SO for imgur.  Cheers :) We know that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\int_0^\infty \frac{\sin x}{x} \, dx=\frac{\pi}{2}.$$ How do I show that $$\int_0^\infty \left\vert\frac{\sin x}{x} \right\vert \, dx$$ converges? Opacity of background-color, but not the text How do I make the cross-browser (including Internet Explorer 6) transparency for the background of a div while the text remains opaque?  I need to do it without using any library such as jQuery, etc. (But if you know of a library that does it I'd love to know so I can look at their code). Is there way to move one vertex and make another move with it proportionally? I'll try to explain in this gif. I am doing this manually in 2 steps - rotate, scale. And I select all vertexes. Is there instrument rotate them in one step or can I move one and make other move too? Show that any set in a metric space can be written as the intersection of open sets Show that any set contained in the metric space $(X, d)$ can be   written as the intersection of open sets.   Definitions: A set $A \subseteq X$ is open if $\forall x \in A$, $\exists \varepsilon&gt;0$ such that $B_{\varepsilon}(x) \subseteq A$. A set $C \subseteq X$ is closed if and only if $X \setminus C$ is open.   I also know that the intersection of a finite collection of open sets is open and that any open ball contained in $X$ is open. How can I prove this question? What are the blue dots next to App names in iOS and macOS? They disappear once you open the apps and then return to the home screen, but I can't tell what they indicate.  Are they apps that haven't been opened since an app update? Or that it thinks have never been opened (not accurate in my case, but that could be a function of a "restore" confusing the phone). Irreducibility of a polynomial if it has no root (Capelli) Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$.   This  to a related question mentions the result is due to Capelli.  I can prove the result if $F$ has characteristic $p$ as follows.  Suppose $f$ is reducible: $f(X)=g(X)h(X)$ with $g(X)$ an irreducible factor of degree $m$, $1\le m&lt;p$. Then if $\alpha$ is a root of $g$ in some extension field $K$ of $F$, we have $$f(X)=X^p-\alpha^p=(X-\alpha)^p$$ so its divisor $g(X)$ must be of the form $(X-\alpha)^m$. Since the coefficient of $X^{m-1}$ in $g$ is in $F$, we have $m\alpha\in F$.  So $\alpha\in F$ because $m$ is invertible modulo $p$.  How would you show the result in other characteristics? Opening PDF in Illustrator breaks up text objects I recently had to convert some PDF files to SVG, and did this by opening the PDF in Illustrator, and saving out to SVG.    The issue was that when Illustrator opened the pdf, many (but not all) text objects would be broken up into different text objects.  For instance, the word "policy" wouldn't be one text object, but rather several text objects, "po", "l", "icy".  There didn't seem to be any rhyme or reason to it.    How can I prevent this from happening? How to change default window background color in Windows 10? Where do I put my systemd unit file? If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t I need to show that $$ \int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx $$ converges if and only if $\alpha + 1 &lt; \beta$. I used substitution and integration to get $$ \lim\limits_{t\to\infty}\int\limits_1^tx^\alpha\sin(x^\beta)dx \overset{u = x^\beta}{\underset{du = \beta x^{\beta-1}}{=} }\lim\limits_{t\to\infty}\left(\frac{1}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1 - \beta}{\beta}}\sin{u}du\right) $$ If $\alpha + 1 -\beta &lt; 0$, then we have a product if a function with bounded anti-derivative and a differentiable decreasing function which tend to 0, and from Dirichlet's criterion, the integral converges.   For the converse, I used integration by parts $$  = \frac{1}{\beta}\lim\limits_{t\to\infty}\left(\left.-u^{\frac{\alpha + 1-\beta}{\beta}}\cos{u}\right|_1^{t^\beta} + \frac{\alpha+1-\beta}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1-2\beta}{\beta}}\cos{u}du\right) $$ For $\alpha + 1 -\beta\geq 0$, I get that the first term indeed diverges, but this is not enough to show total divergence. How can one show it? using gb4e keeping interlinear glosses together on the same page I am using gb4e for several hundred interlinear glosses in my dissertation. Several of the examples span more than one page, that is, the vernacular and gloss will be on one page, while the free translation appears at the top of the next page.   How can I keep all three elements together on the same page? I often have subexamples (a, b, c) that don't need to stay together. That is, 1a can be on one page, and 1b on the next, as long as vernacular-gloss-free translation for each subpart stay together.  In the MWE below, the free translation is at the top of page 2, while the vernacular and gloss are at the bottom of page 1.  \documentclass[11pt,letterpaper]{article} \usepackage{lipsum} \usepackage{gb4e} \noautomath \begin{document} \lipsum[1-4] \begin{exe}     \ex \gll wuz x̃ũch it-ak xohish-em\\              1\textsc{sg.prn} bread eat-\textsc{inf} want.\textsc{prs}-1\textsc{sg}\\         \glt `I want to eat bread.'  \end{exe} \end{document}   A solution was proposed to a similar question (), but it requires me to manually look for what could be dozens of orphans throughout my dissertation (and doesn't work when applied to my example, since it still strands the FT). I would prefer a more elegant solution that takes care of the problem automatically. Why teach linear algebra before abstract algebra? Is there a reason why most undergraduate curriculums put linear algebra before abstract algebra?  I'm asking this because personally it seems to be much easier to understand the architecture behind linear algebra as supposed to simply solving problems after a course in abstract algebra.