Definition talk:Rotation Matrix

Convention?

I was wondering if anyone could describe what is intended to be on the page:

Convention

On the definition page.

I would like to work on a page about 2-D rotation matrices such as

$\mathbf R = \begin{bmatrix}

\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$

and some results of it.

First, please sign your posts.

The book I was ploughing through is wordy and invites an intuitive understanding:

Note that in defining the matrix elements by the equation $a_{i j} \equiv {\mathbf e'}_i \cdot \mathbf e_j \quad (1.7a)$, we have adopted a certain convention. We could just as well have defined matrix elements $a'_{i j}$ by $a'_{i j} \equiv {\mathbf e'}_j \cdot \mathbf e_i = a_{j i} \quad (1.7b)$. Almost all authors use the convention of Eq. $(1.7a)$, but this is an arbirary choice; a completely consistent development of the theory is possible based on the definition $(1.7b)$. In fact, in abstract vector space theory, matrices are usually defined by a convention consistent with Eq. $(1.7b)$ rather than by our rule, Eq. $(1.7a)$. However, by replacing $a_{i j}$ with $a_{j i}$ in Eq. $(1.6)$ and the equations we shall derive presently -- thus interchanging rows and columns or transposing the matrix $R$ -- we may shuttle back an forth between conventions. In chapter $4$ we shall reconsider these issues in a more general setting that permit an easy and complete systematisation.

Incidentally, I sure do hope that what you plan is not "a page" but a series of pages each with one result on it. --prime mover (talk) 07:34, 21 May 2023 (UTC)