The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns by 
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01Or equivalently, the set of rows is orthonormal, and so is the set of columns. TODO proof that it is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.
Connected components of the orthogonal group by Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01The orthogonal group has 2 connected components:
- one with determinant +1, which is itself a subgroup known as the special orthogonal group. These are pure rotations without a reflection.
- the other with determinant -1. This is not a subgroup as it does not contain the origin. It represents rotations with a reflection.
It is instructive to visualize how the looks like in :
- you take the first basis vector and move it to any other. You have therefore two angular parameters.
- you take the second one, and move it to be orthogonal to the first new vector. (you can choose a circle around the first new vector, and so you have another angular parameter.
- at last, for the last one, there are only two choices that are orthogonal to both previous ones, one in each direction. It is this directio, relative to the others, that determines the "has a reflection or not" thing
As a result it is isomorphic to the direct product of the special orthogonal group by the cyclic group of order 2:
A low dimensional example:because you can only do two things: to flip or not to flip the line around zero.
Note that having the determinant plus or minus 1 is not a definition: there are non-orthogonal groups with determinant plus or minus 1. This is just a property. E.g.:has determinant 1, but:so is not orthogonal.
Where most of Ciro Santilli's ancestors came from, and why Ciro has the Italian nationality as well as Brazilian.
More specifically his paternal line comes from Gissi in the Abruzzo region.
Ciro feels really bad by the fact that he does not speak Italian and has never visited Gissi as of 2020.
He would likely be able to learn Italian in like 3 months because it is so similar to Portuguese and French which he already speaks.
And a cycling visit maybe? That would be amazing! en.wikipedia.org/wiki/Giro_d'Abruzzo | www.youtube.com/watch?v=OW7wqa3vNU8&list=UU35qUU5iZPvuzcre43EV8bA&index=25
For what it is worth though, Ciro Santilli does honestly love Europe, and feels a strong desire to make it even awesomer, along with the rest of the world. Despite this being a hopeless attempt due to having more than one natural language is bad for the world.
Members of the orthogonal group.
Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
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