general linear group over a finite field of order . Remember that due to the classification of finite fields, there is one single field for each prime power .
Exactly as over the real numbers, you just put the finite field elements into a matrix, and then take the invertible ones.
This is a good first concrete example of a Lie algebra. Shown at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" has an example.
Every element with this parametrization has determinant 1:Furthermore, any element can be reached, because by independently settting , and , , and can have any value, and once those three are set, is fixed by the determinant.
Remembering that the Lie bracket of a matrix Lie group is really simple, we can then observe the following Lie bracket relations between them:
One key thing to note is that the specific matrices , and are not really fundamental: we could easily have had different matrices if we had chosen any other parametrization of the group.
TODO confirm: however, no matter which parametrization we choose, the Lie bracket relations between the three elements would always be the same, since it is the number of elements, and the definition of the Lie bracket, that is truly fundamental.
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" then calculates the exponential map of the vector as:with:
TODO now the natural question is: can we cover the entire Lie group with this exponential? Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation" explains why not.
Just like for the finite general linear group, the definition of special also works for finite fields, where 1 is the multiplicative identity!
Note that the definition of orthogonal group may not have such a clear finite analogue on the other hand.
The group of all transformations that preserve some bilinear form, notable examples:
- orthogonal group preserves the inner product
- unitary group preserves a Hermitian form
- Lorentz group preserves the Minkowski inner product
We can almost reach the Lie algebra of any isometry group in a single go. For every in the Lie algebra we must have:because has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".
Bibliography:
Mathematical definition that most directly represents this: the orthogonal group is the group of all matrices that preserve the dot product.
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 What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16We looking at the definition the orthogonal group is the group of all matrices that preserve the dot product, we notice that the dot product is one example of positive definite symmetric bilinear form, which in turn can also be represented by a matrix as shown at: Section "Matrix representation of a symmetric bilinear form".
By looking at this more general point of view, we could ask ourselves what happens to the group if instead of the dot product we took a more general bilinear form, e.g.:The answers to those questions are given by the Sylvester's law of inertia at Section "All indefinite orthogonal groups of matrices of equal metric signature are isomorphic".
- : another positive definite symmetric bilinear form such as ?
- what if we drop the positive definite requirement, e.g. ?
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?
Elements of the orthogonal group have determinant plus or minus one by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16 The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Or 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.
Pinned article: Introduction to the OurBigBook Project
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