I have two symmetric matrix 5x5 ,A and B that
A=$\begin{bmatrix}a11&a12&a13&a14&a15\\a12&a22&a23&a24&a25\\a13&a23&a33&a34&a35\\a14&a24&a34&a44&a45\\a15&a25&a35&a45&a55\end{bmatrix}$
B=$\begin{bmatrix}a11&-a12&-a13&-a14&a15\\-a12&a22&a23&a24&-a25\\-a13&a23&a33&a34&-a35\\-a14&a24&a34&a44&-a45\\a15&-a25&-a35&-a45&a55\end{bmatrix}$
and each of matrix is correspond to symmetric angle for
example A is correspond to ‘$α=θ_A$ ‘and B is correspond to ’ $α=θ_B=π-θ_A$’. (for example $θ_A= π/6$ and $θ_B= 5π/6$ )
Now I want to find an Orthogonal Transformation matrix ‘T(α)’ that:
$T^t (θ_A) *A* T(θ_A) = T^t (θ_B) *B* T(θ_B)$
I find this matrix:
T(α)=$\begin{bmatrix}cos(\alpha)&0&0&0&0\\0&-1&0&0&0\\0&0&-1&0&0\\0&0&0&-1&0\\0&0&0&0&cos(\alpha)\end{bmatrix}$
but this is not orthogonal. Could any freind help me please?