How can I approach this linear algebra problem?

Question

Part A) I can only say there is necessarily n real eigenvalues producing n pairwise orthogonal eigenvectors because A is symmetric, but what else is there to say about the eigenvalues utilizing the fact that A has orthonormal columns?

Answer 1

• If $A$ has orthonormal columns, then $A^T A = I$.
• $A$ is also symmetric, we we have $A^2 = I$.
• Now you can check that the eigenvalues must satisfy $\lambda^2 = 1$.