# Why is the special orthogonal group considered the rotation group?

• tensor33
In summary, rotation preserves the orientation of a basis, so the determinant of a rotation must be 1.

#### tensor33

I understand that the special orthogonal group consists of matrices x such that $x\cdot x=I$ and $detx=1$ where I is the identity matrix and det x means the determinant of x. I get why the matrices following the rule $x\cdot x=I$ are matrices involved with rotations because they preserve the dot products of vectors. The part I don't get is why the matrices involved with rotation must have determinant 1.

A pure rotation should not dilate or shrink any volume. The determinant of a linear operator is the scale factor by which the volume element increases. Hence, the determinant of a rotation operator must be 1.

tensor33 said:
I understand that the special orthogonal group consists of matrices x such that $x\cdot x=I$ and $detx=1$ where I is the identity matrix and det x means the determinant of x. I get why the matrices following the rule $x\cdot x=I$ are matrices involved with rotations because they preserve the dot products of vectors. The part I don't get is why the matrices involved with rotation must have determinant 1.

Did you mean $x\cdot x^T=I$??

Muphrid said:
A pure rotation should not dilate or shrink any volume. The determinant of a linear operator is the scale factor by which the volume element increases. Hence, the determinant of a rotation operator must be 1.

That doesn't really explain why the determinant can't be -1, which is what the OP is asking.
The reason is that rotations preserves the orientation of a basis. If we have a right-handed basis, then rotations of this will be right-handed as well.
As an example of an orthogonal matrix that does not preserve the orientation, you can probably take a reflection.

tensor33 said:
I understand that the special orthogonal group consists of matrices x such that $x\cdot x=I$ and $detx=1$ where I is the identity matrix and det x means the determinant of x. I get why the matrices following the rule $x\cdot x=I$ are matrices involved with rotations because they preserve the dot products of vectors. The part I don't get is why the matrices involved with rotation must have determinant 1.
To prove that rotations have determinant 1, you must first define the term "rotation". It's perfectly OK to just take "R is said to be a rotation if R is a member of SO(3)" as the definition, but then there's nothing to prove. Another option is "R is said to be a rotation if R is a member of the largest connected subgroup of O(3)". Then your task would be to prove that the largest connected subgroup is SO(3).

(I actually prefer the terminology that calls members of O(3) rotations, and members of SO(3) proper rotations).

micromass said:
Did you mean $x\cdot x^T=I$??

Ya, I forgot the transpose part.

Thanks to all those who replied, I'm pretty sure I get it now.

## 1. Why is the special orthogonal group important in science and mathematics?

The special orthogonal group is important because it represents the set of all possible rotations in n-dimensional space. This is significant in many fields of science and mathematics, including physics, computer graphics, and robotics.

## 2. How is the special orthogonal group related to the concept of orthogonal matrices?

The special orthogonal group is the group of all n-by-n orthogonal matrices with determinant 1. This means that every rotation matrix is a member of the special orthogonal group, and vice versa. In other words, the special orthogonal group is a subset of the set of all orthogonal matrices.

## 3. Why is the special orthogonal group denoted by SO(n)?

The notation SO(n) comes from the German term "Spezielle Orthogonale Gruppe", which was first used by mathematician Sophus Lie in the 1880s. The "n" refers to the dimension of the space in which the rotations occur.

## 4. Is the special orthogonal group limited to rotations in 3D space?

No, the special orthogonal group can represent rotations in any number of dimensions. The "n" in SO(n) can be any positive integer, indicating the number of dimensions in which the rotations occur. So, SO(3) represents rotations in 3D space, SO(2) represents rotations in 2D space, and so on.

## 5. How is the special orthogonal group different from other rotation groups?

The special orthogonal group is unique because it is the only rotation group that preserves the length of vectors and the orientation of objects. This means that after a rotation is applied, the distance between two points and the angles between lines and planes remain unchanged. Other rotation groups, such as the general orthogonal group and the special unitary group, do not have this property.