Hi!

How do you, properly, verify that a given matrix is a rotation matrix?

Consider the following matrix P:

$\displaystyle

P =

\left[

\begin{array}{c c c c c c c}

1 & 0 & \cdots & \cdots & \cdots & 0 & 0 \\

0 & 0 & \cdots & & & \cdots & \cdots \\

\cdots & \cdots & cos \theta & \cdots & sin \theta & \cdots & \cdots \\

\cdots & \cdots & & & & \cdots & \cdots \\

\cdots & \cdots & -sin \theta & \cdots & cos \theta & \cdots & \cdots \\

\cdots & \cdots & & & & \cdots & \cdots \\

0 & 0 & \cdots & \cdots & \cdots & 0 & 1

\end{array}

\right]

$

where $\displaystyle P_{rr} = P_{ss} = cos \theta, P_{rs} = -P_{sr} = sin \theta, P_{ii} = 1 $ (for $\displaystyle i ~= r, s$) and else $\displaystyle 0$.

It's essentially an identity matrix with that familiar 2x2 rotation matrix setup sort of in the middle there. But mathematically, what conditions must I show are true to establish that this is a rotation matrix? I would think orthogonality is one of them?

Thank you in advance!

(sorry for the missing \cdots, I exceeded the limit of latex code)