I've been trying to prove that if I have $A=[a_{ij}]_{n \times n}$ therefore, if;    Then the matrix is invertible.  Can anyone help me and give me some vocabulary (If I'm not wrong, there is a name for this type of matrix, isn't there?)
I try to find a good proof for invertibility of strictly diagonally dominant matrices (defined by $|m_{ii}|&gt;\sum_{j\ne i}|m_{ij}|$).  There is a proof of this in  but I'm wondering whether there are are better proof such as using determinant, etc to show that the matrix is non singular.
I replaced a non-protected receptacle with a GFCI type and now my circuit tester trips the GFCI as soon as it is plugged in.  The receptacle tested fine before replacement, ground, neutral, hot, etc.  I've double checked the hot/neutral/ground connections.  I've also tried it with the downstream load disconnected.   The GFCI trips the moment I plug in my circuit tester (before pushing the test button).  Otherwise things check out OK.  I can plug in a power tool without it tripping.  Also, the integrated test/reset button work correctly.   I've read that the integrated test button is the only one that truly matters. Is this correct or should I be concerned? I'm sure a home inspector will red flag this if their tester has the same result.  Update  My tester is by "Commercial Electric".  The tester is one of the palm sized three prong type with three leds to indicate the condition of the circuit and a button on top to test GFCI. It's new from the local box store. I have the following file:  ---------- 1 Steve Steve 341 2017-12-21 01:51 myFile.txt   I switched the user to root in the terminal, and I have noticed the following behaviors:   I can read this file and write to it. I can't execute this file. If I set the x bit in the user permissions (---x------) or the group permissions (------x---) or the others permissions (---------x) of the file, then I would be able to execute this file.   Can anyone explain to me or point me to a tutorial that explains all of the rules that apply when the root user is dealing with files and directories? Meaning of backtick in floating-point literal How to select faces by area I found this documentation:   Under "Face Selection Mode" it says:     Face Selection Mode:      ...        Area   Selects all faces that have a similar area as those already selected.   But I can't figure out where in blender this option is hiding? Does installing and using Wine open up your Linux platform to Windows viruses? Just wondering if installing Wine might open up a fairly solid Linux desktop to the world of Windows viruses.  Any confirmed reports about that?  Would you then install a Windows antivirus product under Wine? My machine has a SSD (fairly small) and a Hard disk (large). I want to have windows and a couple of programs on the SSD but I want to have the documents folders (such as my documents and my downloads) on the HD. I also want the default install directory pointing at the larger HD. Can anyone advise or point me to a relevent guide? Thanks. How do I put an image texture on a material in Cycles? I am creating a model of a glass and I would like to put a drawing on it (a png picture).  I know how to do it using Blender Render, but not with Cycles Render. Help :) What does the @ symbol before a variable name mean in C#? Possible Duplicate:          I understand that the @ symbol can be used before a string literal to change how the compiler parses the string.  But what does it mean when a variable name is prefixed with the @ symbol? if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian Prove that if $R$ is a commutative ring in which all the prime ideals are finitely generated, then $R$ is Noetherian.   Here is what I been told to do: Suppose that $R$ is not Noetherian, and use Zorn’s lemma to obtain a maximal element $I$ in the collection of all ideals of $R$ that are not finitely generated.  Then use the following proposition: Let $I$ be an ideal of a commutative ring $R$, and let $r ∈ R$. If the ideals $I +rR$ and {$s ∈ R : sr ∈ I$} are finitely generated, then $I$ is a finitely generated ideal.  Can anyone help please. thanks a lot. Eigenspace of the companion matrix of a monic polynomial How do I prove that the eigenspace of an $n\times n$ companion matrix  $$ C_p=\begin{bmatrix} 0 &amp; 1 &amp; 0 &amp;\cdots &amp; 0\\ 0 &amp; 0 &amp; 1 &amp;\cdots &amp; 0 \\ \vdots&amp;\vdots &amp;\vdots&amp;\ddots&amp;\vdots\\ 0 &amp; 0 &amp; 0 &amp;\cdots &amp;1 \\ -\alpha_0 &amp;-\alpha_1 &amp;-\alpha_2 &amp;\cdots&amp;-\alpha_{n-1} \end{bmatrix} $$ equals $\operatorname{Span}\{v_{\lambda} \} $ where $v_{\lambda}$ is an eigenvector of the companion matrix w.r.t. the eigenvalue $\lambda$: $$ v_{\lambda} =  \begin{bmatrix} 1 \\ \lambda\\ \lambda^{2} \\ \vdots\\ \lambda^{n-1} \end{bmatrix}. $$ Sorry if I keep asking for proof checks. I'll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I'd do a proof that made sense to me and seemed, in some ways, simpler. Trouble is I'm not sure if it's totally correct. It's quite short though. I was just hoping someone could look it over and see if it is a valid proof. Thank you!  Lemma: Every Cauchy sequence is bounded.  Proof: Let $(a_{n})$ be Cauchy. We choose $ 0&lt;\epsilon_{0}$. So $ \forall \; n&gt;m\geq N_{0}$ we have that $\vert a_{n}-a_{m} \vert &lt; \epsilon_{0}$. Therefore $(a_{n})$ is bounded for all $ m \geq N_{0} $ by $ \epsilon_{0} $. Since $ \mathbb{N}_{N_{0}}$ is finite, it is bounded. So, for all $ m&lt;N_{0} $, $ (a_{n})$ is bounded. Therefore $(a_{n})$ is bounded.  I realize I haven't said what the bounds are but I think that's sort of irrelevant. So long as we know it's bounded. Any help is much appreciated! A characterisation of quadratic extensions contained in cyclic extensions of degree 4 Let $D$ be a squarefree integer. I am trying to prove that $\mathbb Q[\sqrt D]$ is contained in a Galois extension of $\mathbb Q$ with Galois group $\mathbb Z/4$ if and only if $D$ is the sum of two squares, $D = a^2 + b^2$ with $a,b\in \mathbb Q$.   The hint in the exercise in Dummit and Foote suggests considering the extension $\mathbb Q[\sqrt{s + s\sqrt D}]$ for the forward direction. However, I am unsure how to show that this extension is Galois, much less that its Galois group is cyclic of order 4.   For the reverse direction, I expressed the given Galois extension as $\mathbb Q[\sqrt{a+b\sqrt D}]$ for some $a,b\in \mathbb Q$. I showed that this extension has $\mathbb Q[\sqrt{c^2 - d^2 D}]$ as a subfield for some $c,d \in \mathbb Q$. From this, I deduced that the $\sqrt D$ and $\sqrt{c^2 - d^2 D}$ generate the same extension, so $\sqrt D = x \sqrt{c^2 - d^2 D}$ for some $x$, and hence that $D = \frac{x^2 c^2}{1 + x^2 d^2} $. It seems that there should be a way to decompose this as a sum of two squares, but I did not see how.   I did not tag this as homework as I am studying it independently. Using url package without hyperref makes link hit boxes erratic I've currently got a url link in a footnote in my document. With the hyperref package, everything works correctly. Because the IEEE PDF Express tool doesn't like having the hyperref package, I've removed it and only left in the url package.  However, some strange behavior occurs as a result of this. Instead of just the link being clickable, entire paragraphs of text before/after end up being clickable and going to the same link. This occurs for both Skim and Preview in Mac OSX.  Does anyone have an inkling of what might be causing this? Dynamically reformatting man pages on terminal dimension changes One common workflow of mine is to open a manual page in a terminal, then another terminal in which to test things. The man page is formatted to the initial dimensions of the first terminal. When I now resize my windows (or have my WM do that for me automatically), there is either a gap to the right of the preformatted page, or lines wrap. At this point I usually q(uit) and !! (run again), which loses my position in the page.  I assume the formatting process is quite CPU intensive, or maybe it stems from ancient times of fixed terminal sizes. The less pager dynamically reacts to terminal resize events, so it should be possible in theory.  I tried perusing man pages, searching the Web, asking on IRC -- the whole lot -- but couldn't come up with anything.   Can I trigger reformatting from within or outside of the man utility? Is there a version of the man utility that resizes the page dynamically?  Is there way to customize some part of the formatting/display process to make it update on SIGWINCH? Display lines in last 10 minutes with specific pattern in logs I need to display lines with Error occurred in last 10 minutes of a log file.   Aug 26 10:50:42 Normal line. Aug 26 10:51:23 Normal line. Aug 26 10:55:33 Error line. Aug 26 10:56:45 Normal line. Aug 26 10:58:12 Error line. Aug 26 11:02:31 Normal line. Aug 26 11:03:32 Normal line. Aug 26 11:04:11 Normal line.   Suppose above sample of log file. I want to display only following two lines  Aug 26 10:55:33 Error line. Aug 26 10:58:12 Error line.   I am using AIX. What's the difference between the atomic and nonatomic attributes? What do atomic and nonatomic mean in property declarations?  @property(nonatomic, retain) UITextField *userName; @property(atomic, retain) UITextField *userName; @property(retain) UITextField *userName;   What is the operational difference between these three? Fusion: Why deuterium and tritium? I would like to know why scientists try to use deuterium and tritium for  and not just the ordinary  ${}^1H$? How to print original variable's name in Python after it was returned from a function? I have enum and use the variables like myEnum.SomeNameA, myEnum.SomeNameB, etc.  When I return one of these variables from a function, can I print their names (such as myEnum.SomeNameA) instead of the value they returned? Confirmation that someone is listening to another person's speech When someone is telling you a very long and detailed story he usually wants to hear some "confirmations" (or response) that you are listening to his story. In Russian we often use something like "tak" (which has a meaning of "ok" and "well, proceed further"), "a-ha" or "uh-huh".  What word serves the same purpose in English and American English? How to sample from a discrete distribution? Assume I have a distribution governing the possible outcome from a single random variable X.  This is something like [0.1, 0.4, 0.2, 0.3] for X being a value of either 1, 2, 3, 4.  Is it possible to sample from this distribution, i.e. generate pseudo random numbers upon each of the possible outcomes given the probability of that outcome. So if I wanted to know what the probability of getting a 2 is, the sampling operation may return 0.34 or something like that.  The reason I ask is that I'm trying to implement an action selection policy for a reinforcement learning method based on a research paper. From what I gather from the paper, the author is able to sample the distribution by "mapping the uniform distribution U[0,1] through cumulative probability density functions obtained by adaptive numerical integration". From this he then samples the transition probabilities for each trial...  I would be grateful for any info on this...  Thanks in advance