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In notes on Lie Theory I'm studying there is a conjugation by (and the remark) "an orthogonal matrix $A$ corresponding to an orthonormal basis compatible with the rotation." Then there is the expression:

$$A\pmatrix{\cos t&-\sin t&0\\\sin t&\cos t&0\\0&0&1}A^{-1}$$

where I can see the matrix in the brackets being conjugated is a rotation.

Is it correct to say the $A^{-1}$ takes a vector that would be rotated to the basis of the rotation matrix. It is then rotated, and the $A$ takes the resultant rotated vector back to its original basis?

Assuming this description is correct, how is $A$ determined?

Thanks

  • This is unclear: How is $A$ determined by what? Is the situation that you have a general rotation $B$ in $SO(3)$ and you want to produce an $A$ for which the expression in the display equation coincides with $B$? – Travis Willse May 16 '17 at 14:26

2 Answers2

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Your description is one correct way of describing it.

In any $3D$ rotation, there are two orthogonal directions that play the role of the $x,y$ axes and a remaining direction which plays the role of the axis of the $z$-axis, i.e. the axis of rotation (note that these three directions are mutually orthogonal). Let's call these axes the "$x$-like axis, the $y$-like axis, and the $z$-like axis" respectively.

$A$ is the transformation that maps the $x$-axis to the $x$-like axis, the $y$-axis to the $y$-like axis, and the $z$-axis to the $z$-like axis.

Ben Grossmann
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This is a way of expressing a rotation about z-axis (let name this coordinate frame with this z-axis as $\{ 1 \}$ ) in a different coordinate frame $\{ 0 \}$.

For vectors described in these two frames we have equation $^{0}v=A^{1}v$, we can calculate $A$ taking orthogonal unit vectors $i,j,k$ and expressing them with the use of their representations in frames as $\{ 0 \}$ and $\{ 1 \}$ as $[ ^0i \ ^0j \ ^0k] =A[ ^1i \ ^1j \ ^1k]$.

Hence $A=[ ^0i \ ^0j \ ^0k]{[ ^1i \ ^1j \ ^1k]}^{-1}$.

Widawensen
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  • Favorit tag: Rotation - love it. With regards, –  May 16 '17 at 14:48
  • @Andrew Thank you, Andrew. I'm collecting things connected with rotations :) – Widawensen May 16 '17 at 14:50
  • If you are interested, here are a few others since I'm "studying" Lie Theory which at this level is primarily matrix Lie groups: rotations, rotations, rotations:https://math.stackexchange.com/questions/2256577/a-linear-algebra-question-relating-to-showing-how-every-element-x-in-so3-is, and in $E(n)$ somewhat related https://math.stackexchange.com/questions/2262658/show-the-set-of-translations-is-a-normal-subgroup-of-the-euclidean-group-en and https://math.stackexchange.com/questions/2267747/why-mathbbt1-is-maximal-in-su2 –  May 16 '17 at 15:18