Eigenvectors of a symmetric 2 2 matrix let p q rn be two

Question

Question: (Eigenvectors of a symmetric 2 × 2 matrix) Let p, q ∈ Rⁿ be two linearly independent vectors, with unit norm (||p||₂ = ||q||₂ = 1). Define the symmetric matrix A ? pq^? + qp^?. In your derivations, it may be useful to use the notation c ? p^?q.

1. Show that p + q and p - q are eigenvectors of A, and determine the corresponding eigenvalues.

2. Determine the null space and rank of A.

3. Find an eigenvalue decomposition of A, in terms of p, q.