Finding an orthogonal Matrix with distinct eigenvalues

Question

The symmetric matrix $A$ below has distinct eigenvalues 12, −6 and −12. Find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $P^TAP=D$. Use the square root symbol '$\sqrt\cdot$' where needed to give an exact value for your answer. $$ \begin{pmatrix} -1 & -5 & -8\\ -5 & -1 & 8\\ -8 & 8 & -4 \end{pmatrix}. $$ I know how to find matrix $D$ but im not sure how to find matrix $P$.

Answer 1

The columns of P are the normalized eigenvectors corresponding to the eigenvalues of A, in the same order that you list the eigenvalues of A in the matrix D.

See https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix for more information