Orthogonal matrices question?

To find Q, just take the columns of P and make them orthonormal. Let u,v,w be the columns of P. Your goal is to find vectors orthonormal vectors u', v', and w', that are themselves linear combinations of u, v, and w. Because A is symmetric, this is possible. To actually do it, you can use what's called the Gram-Schmidt process. To make sure you get the same D, just have the first column of P have the eigenvalue in the first entry of D, the second column of P have the eigenvalue in the (2,2) entry of D, and so on.