Determine an orthogonal amtrix $P$ such that $P^{-1}AP$ is a diagonal matrix, where
$$A= \begin{bmatrix} 7 & 4 & -4 \\ 4 & -8 & -1 \\ -4 & -1 & -8\end{bmatrix}$$
I found the required matrix by using eigen vector approach. But it is not orthogonal.
How to proceed to find orthogonal matrix. Please help
Guide:
Suppose you have found two eigenvectors but they are not orthogonal to each other. $\{ u_1, u_2\}$.
$$Au_1 = \lambda u_1$$
$$Au_2 = \lambda u_2$$
Let $v_1 = u_1$ and $v_2 = u_2 - \frac{v_1.u_2}{\|v_1\|^2}v_1$, then $v_2$ and $v_1$ are orthogonal to each other.
To make them orthonormal, divide by their length.
You might want to check out Gram-Schmidt process.