Does existence of Eigendecomposition of a matrix imply that it is invertible? That is, does the existence of n independent eigenvectors of a matrix imply that the matrix is invertible? If so, what is the reasoning behind eigenvectors and invertibility being connected? It seems intuitively wrong based on what I've learnt.   For instance, suppose A has n independent eigenvectors. Then the eigendecomposition exists:   $$A = PDP^{-1}$$  Then A is invertible?  EDIT: link to another question on this topic:
For any matrix $A$, $AA^T$ is symmetric, and thereby diagonalizable (Spectral Theorem). This implies that $AA^T$ has n linearly independent eigenvectors and therefore full rank.  Does this imply that for any matrix $A$, $AA^T$ is always invertible ? I know there is something wrong here because from the above chain we can conclude that a symmetric matrix is always invertible, which is definitely not true.  Could someone please point out where I am going wrong?
If I create a custom environment based on the equation environment, I get a spacing issue: the next line of text after my environment closes doesn't start on the left margin.  Example code:  \documentclass{article} \begin{document} \newenvironment{myenv}{\begin{equation}}{\end{equation}} \noindent $| \leftarrow$ this is the left margin. \begin{myenv} e ^{i \pi} + 1 = 0 \end{myenv} $| \leftarrow$ this isn't the left margin. \end{document}   Output:    Removing the newline after \end{myenv} solves the spacing problem, but the resulting code is not readable.  Of course, in this example I could just use the equation environment itself. My real issue involves a more complex custom environment that suffers from the same problem, and which also includes an equation environment within it. How to calculate the rotation matrix between 2 3D triangles? I need to calculate the rotation matrix and the translation vector between 2 given triangles in Euclidean space.  This is really just a rotation and a translation, the lengths of sides of the triangle are equal.  Coordinates of the points are real. $P_1, P_2, P_3$ are the 3 points of the 1st triangle, $Q_1, Q_2, Q_3$ are the 3 points of the 2nd.  My idea was: I translate the center of the triangles to the point of origin with $t_1 := \frac{P_1+P_2+P_3}{3}$, $P'_i = P_i-t_1$ and  $t_2 := \frac{Q_1+Q_2+Q_3}{3}$, $Q'_i = Q_i-t_1$ and then I only need to solve the set of 9 linear equations with 9 unknown (the 9 entries of the rotation matrix $R$):  $$ Q'_i = R.P'_i $$ Afterwards I could calculate the translation with  $$ t = t_2 - R.t_1 $$ The problem is: after I translate the triangles to the point of origin the points are linearly dependent ($P_3 = -P_1 - P_2$)  So in fact I only have 6 equation with 9 unknown, which allows multiple solutions for $R$. But I need "one" precise solution, because the triangles are just parts of two identical models, which I need the rotation for.   I think the approach will work, if I use 4 points instead of 3.  But can I somehow calculate the rotation matrix with "only" 3 points? Has the Magical Community breached the Final Frontier? (aka Wizards in Space?) Have they? Given that Muggles put the first man in space in 1961, even by 1991 a lot of Muggleborns perfectly aware of the space-travel would have passed through Hogwarts and into the Ministry, even the Department of Mysteries (which seems to be a bit like their research division/Area 51 combo) And even leaving Muggle technology (which Wizards don't care for or about) aside, Humans have wondered about air travel to &quot;other worlds&quot; and meeting aliens from the time of the Ancient Greeks, surely some Witch or Wizard somewhere might have wondered if it really was possible to fly to the moon? So, is there any canon (aka Books + Pottermore + Rowling interviews) evidence of the Wizarding community having ever travelled or trying to travel to space / other planets or satellites? ETA: Note, I am also totally okay (and in fact would be eager for) historical &quot;Famous Wizard&quot; card-like tidbits from canon-sources.  AWESOME example (that I'm frankly not expecting): When the Muggles managed to land on the Moon, the Department of Mysteries had to scramble to hide the Silver-backed Moon Dragon conservatory they had been maintaining near the landing site. ALSO AWESOME example (that I am hoping for): Warlock Flibberti and Witch Tigget misused this earth-based spell and amusingly found themselves on the moon with their pet monkey instead! Two particles with initial positions and velocities $r_1,v_1$ and $r_2,v_2$ are interacting by the inverse square law (with G=1), so that  $$ {d^2r_1\over dt^2} = - { m_2(r_1-r_2)\over |r_1-r_2|^3} $$ $$ {d^2r_2\over dt^2} = - { m_1(r_2-r_1)\over |r_1-r_2|^3} $$  (the inverse square law along the line of separation). What is the complete solution of these differential equations? What is the position of the two objects as a function of time?  After reading a lot on Wikipedia, I've come to the definition of center of mass and relative coordinates:   $$R(t) ~=~ \frac{m_1 r_1 + m_2 r_2}{m_1+m_2}$$  $$\ddot{r}(t) r(t)^2 ~=~ (m_1+m_2)G$$  Where $R$ is the center of mass, and $r$ is the displacement between the particles... Is this correct? How do I proceed to solve the differential equation? Need help in the longtable with the figure I'm trying to create a bowling animation and I want the ball to hit the pins. I already added the Rigid Body physics to the objects but can't get the ball to move.  Obviously I could just add keyframes but   that seems to interfere with the Rigid Body physics it doesn't look too realistic.   Is there a way to add some kind of motion to the ball? What is the exposure triangle? How do the "sides" affect my photographs? Problems with \printbibliography I am writing my thesis and need to do this with apa citation style.  I used the following code in my main document:  \usepackage[backend=biber,style=apa,sorting=none,natbib=true]{biblatex} \addbibresource{library.bib}  \usepackage[autostyle=true]{csquotes}    and in the end of my main document..  \printbibliography[heading=bibintoc,[title={References}]   However I always get this error:     "Undefined control sequence.   \printbibliography[heading=bibintoc,[title={References}]"   Furthermore in my references table, I get (yearmonthday) instead of (2015)e.g. and the references are not alphabetically sorted. Can my friend and I play minecraft together with different mods? I'm wondering if I could play Minecraft with my friend. A modded Minecraft, only myself with mods and him without mods.  Is it possible? I have a bunch of external and internal HDDs that I use on a Linux system. I only have Linux systems, so using a Linux file-system would only make sense, right? However I'm currently using NTFS everywhere, because it gives me the most usable space out of HDDs.  I would like to switch to Linux file-systems now though, mostly because of permissions and compability (e.g. I can't get my LUKS encrypted NTFS partition to resize under Linux, keeps telling me to chkdsk under Windows).  However when I formatted those HDDs I tried out a bunch of different filesystems and every Linux filesystem, even ext2 which as far as I know has no journaling, used a lot of space for itself. I don't recall exact values, but it was over 100GB that NTFS got me more on a 2TB HDD, which is a lot.  So my question is: Is there a way to make ext-filesystems use less space for themselves? Or is there another filesystem (I've tried ext2, ext3, ext4, NTFS and vfat - None of them came even close to the usable space NTFS offered me) with perfect Linux support and great usable space?  I'd love to hear about how and why filesystems (especially ext2 which has no journaling) use that much more space than NTFS and I don't know where else to ask. I'd mostly prefer a way to use ext4 without journaling and anything else that uses up this much space, if that's possible. How to do one-class text classification? I have to deal with a text classification problem. A web crawler crawls webpages of a certain domain and for each webpage I want to find out whether it belongs to only one specific class or not. That is, if I call this class Positive, each crawled webpage belongs either to class Positive or to class Non-Positive.  I already have a large training set of webpages for class Positive. But how to create a training set for class Non-Positive which is as representative as possible? I mean, I could basically use each and everything for that class. Can I just collect some arbitrary pages that definitely do not belong to class Positive? I'm sure the performance of a text classification algorithm (I prefer to make use of a Naive Bayes algorithm) highly depends on which webpages I choose for class Non-Positive.  So what shall I do? Can somebody please give me an advice? Thank you very much! $\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$ Let $R$ be a commutative ring, $a_1, ..., a_n$ its elements and  $\phi: R[x_1, ..., x_n] \to R$  defined by  $ \phi(f(x_1, ..., x_n)) = f(a_1, ... ,a_n)$  a ring homomorphism.  Prove: $\ker \phi = (x_1-a_1, ..., x_n-a_n)$  It is obvious that $(x_1 - a_1, ..., x_n -a_n) \subseteq \ker \phi$. I'm not sure how to prove the converse.  At this point I don't know any division algorithms for multivariable polynomials, only for the ones in $R[x]$(and the book from where I taken the exercise doesn't assume the reader to know something beyond basics of rings and ideals, and the division algorithm for $R[x]$). Though I know this could be solved for $i = 1$ by dividing by $x-a$:   Let $f(x) \in \ker \phi$, divide by $x-a: f(x) = q(x)(x-a) + r, f(a) = r = 0$, so $f(x) = q(x)(x-a) \in (x-a)$. I am running out of storage, I usually have bought a lot of my games on disk but if I buy a game on the store then delete it, and want to redownload it do I have to buy it again Positrons versus holes as positive charge carriers : [The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932.]  and:  [Dirac's idea is completely correct in the context of solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge — the charge of the ionic lattice — that cancels out the electric charge of the sea.]  It always confused me to think of holes as positive charge carriers in semi-conductors as not being real: real electrons move from one lattice-position to another lattice-position, which effectively looks like a positive hole in the lattice that is moving in the other direction, but in reality a real electron moves, the hole is kind of an "illusion".  On the other hand the positrons are always introduced as real hard-core particles.   The quotes from the Wikipedia article make me unsure: how should I look upon these phenomena?   Edit: holes in a Dirac sea give rise to real pos. entities in one case and to unreal pos. entities in another - how can we distinguish, is it a matter of formalism? Can someone tell me if $\frac{3}{5}$, $2+3\sqrt{-5}$, $\frac{3+8\sqrt{-5}}{2}$, $\frac{3+8\sqrt{-5}}{5}$, $i\sqrt{-5}$ are all quadratic integers in $\mathbb Q[\sqrt{-5}]$. And if so why are they in $\mathbb Q[\sqrt{-5}]$. Today I started having the following issues in Pokémon GO:   I can't spin pokéStops and get rewards When I catch pokémon, even the lowest CP ones, they escape every time. I've tried using razz berries, that doesn't work. Neither does using great balls.   I do have Internet. I'm not on wifi. I can send messages and browse Internet.  NOTE: I did try to modify my gps location and I was unsuccessful. I only got a pokémon or two and deleted them immediately. I do apologize for trying to hack it but I'm now fully legit. How do I restrict my kids' computing time? Access to our computer (not only to the internet) needs to be restricted for the accounts of my kids (7, 8) until they are old enough to manage this by themselves. Until then we need to be able to define the following:   the hours of the day when computing is o.k. (e.g. 5 - 9 pm) the days of the week when computing is not o.k. (e.g. mondays to fridays) the amount of time allowed per day (e.g. 2 hours)   In 11.10 all of the following that used to do the job don't work any more:   : for > 11.10 no more available through the ppa. : command line alternative, but from 11.10 removed from the repositories. : Looks great but repeatedly crashes to force restarting X-server. So we can't use or recommed this program at the moment.   Are there any other alternatives? Set myself as owner of /etc with chown command now getting all kinds of errors I needed to edit a .config file (which I know for a fact is not creating these problems, just to clear that up) and it wouldn't let me save it, so I took ownership of /etc and all contents with the command chown -hR username /etc and that let me edit the .config file but now whenever I try to install any packages or use any sudo commands, it doesn't work  (I've checked for errors in the sudo file in sudoers.d and there was nothing wrong with that).   So how do I return it to the previous owner? Lightning Components Basics: Attributes and Expressions Trail The error :      The Packed field is either not using the lightning input component,   the checked value of the item's Packed__c attribute or the correct   type of input.   My Code:  &lt;aura:component &gt;     &lt;aura:attribute name="item" type="Camping_Item__c"                      required="true"                     default="{Name:'Tent', Price__c:100, Quantity__c:1, Packed__c:true}"                     /&gt;         &lt;p&gt; The Item is &lt;ui:outputText value ="{!v.item}"&gt;&lt;/ui:outputText&gt;&lt;/p&gt;          &lt;ui:outputText value="{!v.item.Name}"/&gt;     &lt;lightning:input type="toggle" label="Packed" name="togglevalue" value="{!v.item.Packed__c}" /&gt;     &lt;lightning:formattedNumber value="{!v.item.Price__c}" style="currency" currencyCode="USD" currencyDisplayAs="symbol"/&gt;      &lt;lightning:formattedNumber  value="{!v.item.Quantity__c}"/&gt; &lt;/aura:component&gt; I'm studying molecular biology and I'm trying to understand an experiment which shows the importance of promoters in the relative transcription level (RT). The image below comes from Rolf Knippers' book "Molekulare Genetik" (8th edition).    The legend says (among other things):     Die erste Zeile gibt die normale "Wildtyp"-Sequenz der 5'-flankierenden Region wieder.   Which, in english, means something like:     The first line repeats the normal wild-type sequence of the 5' flanking region.   The column on the right gives the relative transcription (RT) level, 1.0 being the highest possible level of transcription. As we can see, the lines where some parts of the "5' region" have been deleted give quite low RT levels, since some regions of the promoter are missing.  My questions are the following:  1) According to  and , I understand that RNA polymerase reads and uses both coding and non-coding strands in order to synthesize RNA. Therefore, how did they manage to use the sequences strands regions had been deleted to make a polymerase "read them" and perform transcription?  2) If the polymerase reads the template strand in the 3' -> 5' sense, shouldn't we talk of "ATAT box" or "TAAC box" instead of "TATA" or "CAAT" boxes? Does it mean that the "promoter" regions they are using on the graphic above are actually on the coding strand?  Thanks a lot for your help.