Non-invertible are excluded "because" otherwise it would not form a group (every element must have an inverse). This is therefore the largest possible group under matrix multiplication, other matrix multiplication groups being subgroups of it.
So dominant that it is usualy called just "zip".
In simple terms, represents keystrokes of a piano, but it can likely also represent other effects (TODO confirm: bend, vibrato, slides, attack strength)
Can contain multiple parallel tracks as seen from the Wikipedia example: en.wikipedia.org/wiki/File:MIDI_sample.mid
You can see what it contains well with GUI music editor.
To actually listen to MIDI, you need a software synthesizer, which knows what sound to make for each note. One way to specify such instruments is the SoundFont file format.
Contains instructions on how to synthesize MIDI.
Extension: sf2.
Can be imported for example by:
- vmpk
- Csound with
sfload: www.csounds.com/manual/html/sfload.html
Opens a virtual MIDI piano GUI. It just works on Ubuntu 20.04: askubuntu.com/questions/34391/virtual-midi-piano-keyboard-setup/1298026#1298026
VMPK is a virtual device that replicates what you would get by connecting a physical MIDI keyboard to your computer. It is not a software synthesizer on its own. But it does connect to a working synthesizer by default (Sonivox EAS) which makes it produce sounds out-of-the box.
TODO: then I messed with my sound settings, and then it stopped working by default on the default "MIDI Connection" > "MIDI Out Driver" > "Network". But it still works on "SonivoxEAS".
A hello world of actually connecting it to a specific software synthesizer manually on Advanced Linux Sound Architecture with
aconnect can be found at: askubuntu.com/questions/34391/virtual-midi-piano-keyboard-setup/1298026#1298026Save to a MIDI file: askubuntu.com/questions/709673/save-as-midi-when-playing-from-vmpk-qsynth/1298231#1298231
Reasonable default key mappings to keyboard covering 2 octaves.
3 multiple simultaneous keys did not work (tested "ZQI"). This might just be a limitation of my keyboard however.
TODO how to save to a
.mid file? askubuntu.com/questions/709673/save-as-midi-when-playing-from-vmpk-qsynthThe generic tool recommendation question: unix.stackexchange.com/questions/7425/is-there-a-robust-command-line-tool-for-processing-csv-files
The orthogonal group is the group of all matrices that preserve the dot product by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16 The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Let's show that this definition is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.
Note that:and for that to be true for all possible and then we must have:i.e. the matrix inverse is equal to the transpose.
These matricese are called the orthogonal matrices.
TODO is there any more intuitive way to think about this?
Lots of features, but slow because written in Python. A faster version may be csvtools. Also some annoyances like obtuse header handing and missing features like grep + cut in one go: csvgrep and select column in csvkit.
Simple example:output:
printf '00,11,22\n33,44,55\n' | csvgrep -H -c2 -r '^11$' | tail -n+200,11,22There seems to be no way without a pipe, you seem to need to reparse the columns, e.g. the tutorial at: csvkit.readthedocs.io/en/latest/tutorial/2_examining_the_data.html#csvgrep-find-the-data-you-need does:
csvcut -c county,item_name,total_cost data.csv | csvgrep -c county -m LANCASTERA compiled executable under
/usr/bin/csvtool, has an Ubuntu 23.04 package: manpages.ubuntu.com/manpages/lunar/en/man1/csvtool.1.htmlThere seems to be no sane filtering mechanism however: stackoverflow.com/questions/46540752/using-csvtool-call-to-filter-csv-in-bash
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact