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

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Tom Mattson

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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 [itex]\theta[/itex] about the [itex]z[/itex] axis is described by the following matrix:

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

It's a piece of cake to show that [itex]det(R_z(\theta))=1[/itex].

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

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

It's a piece of cake to show that [itex]det(R_z(\theta))=1[/itex].

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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.

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lurflurf

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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

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