Rotation of Matrices and their interpretation

Given are now two matrices and I have to discuss what the given functions are doing (geometrically). Maybe you can revise/add the following:

given are the matrices

$ A = \begin{pmatrix} \cos(a) & -\sin(a) & 0 \\ \sin(a) & \cos(b) & 0 \\ 0 & 0 & 1 \end{pmatrix}$

$ B = \begin{pmatrix} \cos(b) & 0 & -\sin(b) \\ 0 & 1 & 0 \\ \sin(b) & 0 & \cos(b) \end{pmatrix}$.

(1) What does $f: \mathbb{R^3} \to \mathbb{R^3}, x \to Ax$ mean?

It means a rotation around the z-axis.

I do $A*(1,0,0) = (\cos(a), \sin(a), 0)$. This is a rotation around z-axis counterclockwise.

I do $A*(0,1,1) = (-\sin(a), \cos(a), 0)$. This is a rotation around z-axis counterclockwise.

I do $A*(0,0,1) = (0,0,1)$. It remains.

(2) What does $g: \mathbb{R^3} \to \mathbb{R^3}, x \to Bx$ mean?

It means a rotation around y-axis.

I do $B*(1,0,0) = (\cos(b), 0, \sin(b))$. This is a rotation around y-axis clockwise.

I do $B*(0,1,0) = (0, 1, 0)$. It remains.

I do $B*(0,0,1) = (-\sin(b), 0, \cos(b))$. This is a rotation around y-axis clockwise.

(3) What does $h: \mathbb{R^3} \to \mathbb{R^3}, x \to ABx$ mean?

I do A*B and then

I do $AB*(1,0,0) = \begin{pmatrix} \cos(a)cos(b) \\ \sin(a)\cos(b) \\ \sin(b) \end{pmatrix}$. This is.. ?!

I do $AB*(0,1,0) = \begin{pmatrix} -\sin(a) \\ \cos(a) \\ 0 \end{pmatrix}$. So this should be again a rotation around z-axis for (a). Regardless of what degree b is!

I do $AB*(0,0,1) = \begin{pmatrix} -\cos(a)\sin(b) \\ -\sin(a)\sin(b) \\ \cos(b) \end{pmatrix}$. ?!

And so on.

Solution 1:

I think the answer to your question is Euler's rotation theorem which states that the composition of two rotations in $3$-d space is also a rotation. Thus, the rotations have the structure of a group.

For example, in your case, composing rotation $A$ with $B$ is another rotation along a different axis by this theorem.

Solution 2:

A rotation in 3-dimensional space is specified by two pieces of data: its axis and amount (angle) of rotation. To find out which is the axis for composition of two rotations A and B, find the eigenspace of eigenvalue 1:. That is Solve the system $(AB-I)v=0$.

Then the angle is $\cos^{-1}\frac12( tr(AB)-1) )$