Let $A \in {M_n}$ and $\left| {{a_{ii}}} \right| &gt; \sum\limits_{j \ne i} {\left| {{a_{ij}}} \right|} $.Why does $A$ is nonsingular?
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.
Remove empty elements from an array in Javascript How do I remove empty elements from an array in JavaScript?   Is there a straightforward way, or do I need to loop through it and remove them manually? Today while troubleshooting a service broker problem, I discovered that the database owner was the Windows login of an employee who had left the company. His login had been removed and thus the query notifications were failing.   Supposedly the best practice for dealing with this is to make 'sa' the database owner. We changed it and that cleared out the queue.   My (very elementary) question: what is the database owner and what is its purpose? Well, just another idea came up into my mind and i have no idea how to solve it :D Is there infinitely many prime numbers, which are not repunits and their inverse is also prime? (For example, inverse of 31 is 13 which is also prime. i didn't have any other name to describe the function!) P.S:I now know they are called s. Quantum entanglement faster than speed of light? In Minecraft, the NBT (Named Binary Tag) format is used to store data of our Minecraft worlds, including block, entity, and item properties. In the Java Edition of Minecraft, you can modify this data by including the data in the command, like so: /give @s diamond_sword{Enchantments:[{id:&quot;minecraft:knockback&quot;,lvl:10s}]}  In the Bedrock Edition, on the other hand, NBT is inaccessible from commands. In order to modify NBT, we need to use an external editor. What are some useful item properties that I can modify with an NBT editor, and how can I do so?  Related: What's needed for a complete backup system? For my new server, I want to setup a proper backup solution. I've found a great setup that will do twice-daily incremental backups via Dropbox. I plan on backing up my various databases, the webroot directory, the /etc directory/repository, and /var/log.   What else do I need to know to do a proper backup, and what is the standard setup here to ensure you can quickly restore from a backup in the case of a system failure?  I'm thinking of using Puppet, as it describes how the system should be. My restore procedure would look like this:   Install Puppet Run my puppet config Restore my backups from Dropbox (Should I create a script to do this? Probably)   This should also let me create a clone of my production server for use in dev environments, correct? Am I missing anything of importance? It is a common practice to apply PCA (principal component analysis) before a clustering algorithm (such as k-means). It is believed that it improves the clustering results in practice (noise reduction).   However I am interested in a comparative and in-depth study of the relationship between PCA and k-means. For example, Chris Ding and Xiaofeng He, 2004,  showed that "principal components are the continuous solutions to the discrete cluster membership indicators for K-means clustering". However, I have hard time understanding this paper, and Wikipedia actually .  Also, the results of the two methods are somewhat different in the sense that PCA helps to reduce the number of "features" while preserving the variance, whereas clustering reduces the number of "data-points" by summarizing several points by their expectations/means (in the case of k-means). So if the dataset consists in $N$ points with $T$ features each, PCA aims at compressing the $T$ features whereas clustering aims at compressing the $N$ data-points.  I am looking for a layman explanation of the relations between these two techniques + some more technical papers relating the two techniques. Possibility of recovering files from a dd zero-filled hard disk I have "zero filled" (complete wiped) an external hard disk using dd, and from what I have heard: people said you should at least "zero fill" 3 times to be sure that the data are really wiped and no one can recover anything.  So I decided to scan the disk once again after I've zero filled the disk. I was expecting the disk to still have some random binary left. It turned out that it has only a few sequential bytes in the very beginning. This is probably the file structure type and other headers stuff. Other than that, it's all zeros and nothing else.  So if we have to recover any file from a zero filled disk, ...how? From what I've heard, even you zero fill the disk, you should still have some data left. ...or could dd really completely annihilate all data? Finding the missing digits of $23!$ Can I recover my deleted world? I was playing terraria today, and my friend came and deleted the my world.  Is there a way for me to recover my world? How do you deserialize json properties that are reserved words in apex? My furnace pilot light keeps going out so I changed the thermocouple. Now the furnace will come on and heat up to the degree set on my thermostat, then the pilot light will go out and the furnace will not come back on.    Could the thermostat be bad and cause the pilot light to go out? How to use IDM on Mozilla Firefox Quantum I want to use Internet Download Manager for downloading on Firefox. According to this  it can be done with the extension called Flashgot. But Flashgot does not work on new Firefox Quantum. However, I found  that can be used to interrupt browser's downloading jobs by redirecting these jobs to an external download manager like Internet Download Manager (IDM). But I don't know how to integrate IDM with this new extension. I wrote IDM's location (/.wine/drive_c/Program Files (x86)/Internet Download Manager/IDMan.exe) in the box that says "Path to the executable" but when I try to download something, an error is appearing.     The preference page of the extension        The error that is appearing while trying to download files     How can I make it work? Thanks beforehand! I have the following page (deadlink: http://www.workingstorage.com/Sample.htm ) that has a footer which I can't make sit at the bottom of the page. I want the footer to  stick to the window bottom when the page is short and the screen is not filled, and stay at the document end and move down as normal when there is more than a screenful of content (instead of overlapping the content).  The CSS is inherited and befuddles me. I can't seem to change it properly to put a minimum height on the content or make the footer go to the bottom. For which values $r, s \in \mathbb R$ is the set $\{u, v, w\}$ linearly independent? Is clockwise or counter-clockwise rotated text easier to read? When rotating text with css, should it be rotated clockwise or counter-clockwise?  Is one more readable than the other?  Counter Clockwise:   &nbsp;&nbsp;&nbsp;&nbsp;&mdash;&nbsp;&nbsp;&nbsp; Clockwise: Some one who is not good at keeping secrets. In my native language it is called "chugalkhor" but it's a slang. So I can't translate it.  What do you call such a person who can't keep secrets because his instincts don't let him. Whoever or whomever: 'happy for ___ has the pleasure of working with you next.' So sad to lose you, yet happy for whomever has the pleasure of working with you next. Avoiding jumping frames in beamer Sometimes I want an element on a frame to change in steps. I do this by  \only&lt;2&gt;{   ...   }   \only&lt;3&gt;{   ...   }     etc.  But this causes the frame to jump due to different sizes of the included content. How can this be avoided? Word for "not consciously aware of" What's the word to convey the meaning "not consciously aware of"?