In Linear regression, we have $\textbf{$\theta$} = \textbf{$\left (X^TX\right )^{-1} X^T y$}$.   In Ridge regression, we have $\textbf{$\theta$} = \textbf{$\left ( \lambda I+ X^TX\right )^{-1} X^T y$}$.  I learnt somewhere that while $\textbf{$X^TX$}$ is not guaranteed to be invertible, $\textbf{$\lambda I + X^TX$}$ is guaranteed to be invertible.  Is this true? If so, why?
How to prove a general matrix invertible given by as below?  How to prove that $A^TA+I$ is always invertible for $\forall A \in \mathbb{R}^{n\times n}$?
Can you filter out certain SE sites from the hot questions list? Is it possible to filter out Stack Exchange sites from the hot questions list? Word to describe the way cattle and buffalo consume fodder? If we allow that the energy to allow quantum tunnelling through a potential barrier has been borrowed by the Heisenberg energy-time uncertainty relation, from the vacuum energy. How is the borrowed energy paid back? I've been getting this undefined symbol building with this command line:  $ gcc test.cpp Undefined symbols:   "___gxx_personality_v0", referenced from:   etc...   test.cpp is simple and should build fine.  What is the deal? Travelling as unmarried couple in Dubai I'd wish to visit Dubai with my girlfriend, but after reading  and  I started to get a little worried about it. How should I book an hotel? Must I book two singles? Could it be a problem if we walk together in the city and go together to restaurants and malls without being married? Is WUBI freely available? I checked the link to download WUBI and the only way to proceed is by giving some amount as donation. I will surely do it, but I don't have a Paypal account and I am in urgent need to install Ubuntu. Can someone help me? Complete Installation Guide for Android SDK / ADT Bundle on Ubuntu What all needs to be done to install Android SDK / ADT Bundle on Ubuntu?  Each of the steps in the answer was found on some different forum as I encountered obstacles getting started with my Android SDK and posted questions in appropriate forums thereafter. So I thought this would be useful in that it is a one stop solution available, in case one ever needs to re-install Ubuntu and install the Android SDK / ADT Bundle from scratch! A zero-dimensional ring is Noetherian? Proposition: Let $A$ be a non-zero ring that is not a field. Suppose $A$ is zero dimensional. Then it is Noetherian.   Proof: Let $p$ be a prime ideal of $A$. If $p$ is not maximal, then $p \subsetneq m$ for a maximal ideal $m$. Hence $dim A \ge 1$, contradiction. Hence every prime ideal is maximal. Consequently, every increasing (or decreasing) chain of prime ideals will be trivial in the sense that it will have length $0$. Hence the ring is Noetherian (and Artinian).  Is there a problem with the above proof? Arrows describing matrix Lets say I have a 5x5 matrix. I want to place 2 leftrightarrows (with descriptions "3", "2") underneath the matrix, but really close to it: first pointing from col1 to col3 and second from col4 to col5. Plus updownarrows with similar outcome. How can I achieve that?  draft:  1 1 1 | 1 1  &lt;--> &lt;-> Where should the period be put when an entire sentence is quoted at the end of a sentence? How can time dilation be symmetric? (already output) in Debug logs I am trying to add a list into an another nested list before adding values of a list to nested list it is getting the correct value but after adding i am getting (already output) in debug log when i checked.    public List&lt;String&gt; fieldName {get;set;}  public List&lt;List&lt;String&gt;&gt; fileValuesXML; // To get the field values of xml  for(Dom.XMLNode child : SalesforceObjecet.getChildElements()) {                        objecetName = child.getText();             system.debug('Object Name in XML field  ' + objecetName );              for(Dom.XMLNode subchild : child.getChildElements())              {                 system.debug('Sub child ' + subchild.getText() );                 for(Dom.XMLNode subchildfield : subchild.getChildElements())                 {                     system.debug('Sub child Field ' + subchildfield );                     fName  = subchildfield.getText();                     fieldName.add(fName);                     //fileValuesXML.add(fieldName);                     //system.debug('Filed Name in XML field  ' + fieldName);                 }                  system.debug('Filed Name before affing to filevalueXML  ' + fieldName);                                     fileValuesXML.add(fieldName);                 fieldName.clear();                 system.debug('Filed Name after affing to filevalueXML  ' + fileValuesXML);             }    }    I am getting values in debug log as:     Filed Name before affing to filevalueXML  (AccountSource, field3)   Filed Name after affing to filevalueXML  ((), (already output),   (already output)) Failed to Login : Invalid session (Try restarting your game) Minecraft Error When I try to connect to servers or join up on lan worlds in Minecraft it says "Failed to Login : Invalid session (Try restarting your game)". Multiplayer used to work fine but now keeps coming up with this error. Can anybody help? Diagnostics for logistic regression? For linear regression, we can check the diagnostic plots (residuals plots, Normal QQ plots, etc) to check if the assumptions of linear regression are violated.  For logistic regression, I am having trouble finding resources that explain how to diagnose the logistic regression model fit. Digging up some course notes for GLM, it simply states that checking the residuals is not helpful for performing diagnosis for a logistic regression fit.  Looking around the internet, there also seems to be various "diagnosis" procedures, such as checking the model deviance and performing chi-squared tests, but other sources state that this is inappropriate, and that you should perform a Hosmer-Lemeshow goodness of fit test. Then I find other sources that state that this test may be highly dependent on the actual groupings and cut-off values (may not be reliable).  So how should one diagnose the logistic regression fit? Compute $P(X_1+\cdots+X_k\lt 1)$ for $(X_i)$ i.i.d. uniform on $(0,1)$ Consider $N=\min\{n: S_n&gt;1\}$, where $S_n=X_1+\cdots+X_n$ and $(X_i)_{i=1}^\infty$ is i.i.d. uniform on $(0,1)$. So, $N$ is the first time that $(S_n)_{n=1}^\infty$ crosses $1$. I'd like to calculate $E(N)$. To this end I'd like to calculate $\Pr(N\gt k)=\Pr(S_k\lt 1)$.  Is there a way to actually do this for any arbitrary $k$? I mean, certainly there is as I know about the distribution of $S_n$ but the formula is just formidable when you move with $k$ toward $\infty$ and even though tractable with a computer, I don't think it's tractable without it.  Is there maybe a better way to calculate the expectation? Thanks for any help. Finding files by only knowing extension? \hfill \break does not work Possible Duplicate:           My name is Greg — this is Greg 's post.  If my name ended with an 's', I am aware of the proper apostrophe usage (James&nbsp;→ James'), but how should this be pronounced?  Phonetically, am I Jameses best friend or James best friend? I'm writing a technical documentation and I want to explain step-by-step how to go through various menus.  For example I used to write Open with &gt; VLC media player.  That's how it looks in Windows 8:     What I want in Latex should looks similar to this: light grey box, dark grey border and black windows-like font.    I've seen , suggesting the package  - but it looks very old fashioned - are there any more modern approaches? When using the ssh or ftp commands from the Bash shell, does the server that I am connecting to learn of the domain name used? I understand that the domain name is locally translated into an IP address via DNS. In HTTP, after that happens, the server is told the original domain name as well in order to serve the correct page, or to present the correct TLS cert (SNI).  host serverfault.com GET /   Does a similar phenomenon happen when connecting to ssh or ftp?  I ask because I am  trying to ssh into a server (GoDaddy webhosting) which expects a domain name, but is not letting me in when I try to connect via user@IPaddress as the DNS is not yet moved to the GoDaddy IP address.