Hi!

Assume $\displaystyle B = P^TAP$, where $\displaystyle P$ is the rotation matrix (below) with angle $\displaystyle \theta$ and position $\displaystyle r, s$. Show that $\displaystyle B$ is symmetric when $\displaystyle A$ is, and that we then also have: $\displaystyle B_{rs} = B_{sr} = sin\theta cos\theta (A_{rr} - A{ss}) + (cos^2\theta - sin^2 \theta) A_{rs}$.

The rotation matrix:

$\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 $\displaystyle 0$.

I don't know where to start!

Help is greatly appreciated!