Prove Determinant of Rotation Matrix is 1 w/Continuity Argument

[asdf60]

What's a continuity argument? For example, a question asks to prove that the determinant of a rotation matrix is always 1 using a continuity argument?

 

[asdf60]

Anyone? How would i prove in general that the det of a rotation matrix is 1?

 

[quantumdude]

I am unsure of what you mean by a continuity argument as it pertains to matrices and determinants. But you can straightforwardly show that the determinant of a rotation matrix is 1 by writing down the matrix and taking its determinant.

For instance, a clockwise rotation by an angle $\theta$ about the $z$ axis is described by the following matrix:

$$R_z(\theta)=\left[\begin{array}{ccc}\cos(\theta)&sin(\theta)&0 \\ -\sin(\theta)&\cos(\theta)&0 \\ 0&0&1 \end{array}\right ]$$

It's a piece of cake to show that $det(R_z(\theta))=1$.

 

[asdf60]

I'm not sure what the question means by continuity argument either. That's why I asked.

I know could just take the determinant of the rotation matrix. Since by euler's rotation theorem any rotation matrix M can be expressed as rotations over three perpendicular axis, det M = det R1 * det R2 * det R3 = 1, where R1, R2, and R3, are just the general rotation matrices for rotating over x,y, and z axis. But I wasn't sure if this 'proof' is sufficiently rigorous and general.

 

[lurflurf]

When asking questions like this one it would be nice to define what you are talking about. In this case a rotation matrix. The definition
Rotation matrix: A matrix for which det(A)=1
would be very helpful

Another definition is
Rotation matrix: An isometric orientation preserving linear transform
by isometric I mean the inner product defined by A matches the one defined by I (idenity matrix)
(Ax)'Ay=x'y
where ' is the adjoint (conjugate transpose or if A is real transpose)
immediately we have
A'A=I
This only insures abs(det(A))=1
so we turn to the orientation preserving bit
That is fancy talk, but it means that I is our prototype rotation
that is
T(x,y,z)=(-x,y,z)
would be a "bad" or improper rotation
we want for A a rotation and B any matrix
det(AB)=det(B)
which again does not teach us much
I said we want I to be a prototype
det(I)=1 but again we want to go further
say h is a small number and we wish to construct an almost rotation
I+Ah
we have
(I+Ah)'=I+A'h
and
(I+Ah)^-1~I-Ah
thus we require
A'=-A
we could also make better almost rotations
I+Ah+A^2h^2/2+A^3h^3/6+...
The ultimate result being the actual rotation
exp(Ah)
and
of course
A'=-A->tr(A)=0
det(exp(At))=exp(tr(At))=exp(0)=1
Thus rotation matricies are of the form exp(A) where A'=-A thus have det=1