Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof resit / 6 / a / i

The characteristic polynomial isThus the eigenvalues are 1 and 11. Corresponding eigenvectors are $(3,-1{)}^T$ and $(1,3{)}^T$. Their dot product is zero and each has Euclidean norm $\sqrt{10}$. Normalizing givesThe columns are orthonormal, so $P^{T}P=I$. The eigenvector equations give $AP=PD$ and hence $P^{T}AP=D$, proving the claimed orthogonal diagonalization of a real symmetric matrix.