Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix}$, how would I find a real orthogonal matrix $P$ such that $PAP^t$ is a diagonal matrix?
I've found the eigenvalues $0, \dfrac{9\pm\sqrt{105}}{2}$, but I don't know how to proceed from here.
There exists a basis of eigenvectors since $A$ is symmetric. Form an orthonormal basis of eigenvectors. Form the matrix whose columns are the basis vectors, as your $P$.