Orthogonality, Least Squares, and Spectral Clinic

Complete P11–P15. State whether your conclusions use real or complex scalars.

P11. Missing direction. Show that a 90-degree real rotation has no real eigenvector. For diag(−3,1), explain the effect of starting power iteration at (1,1) versus (0,1). Separate sign changes, eigenpair accuracy and dominance.

P12. Spectral boundary. Prove [[1,1],[0,1]]^k=[[1,k],[0,1]] by induction. Why does spectral radius one not guarantee bounded powers? Derive powers of [[0.5,10],[0,0.5]] and describe transient growth followed by decay.

P13. Positive structure. Prove [[2,−1],[−1,2]] is PD using leading minors and complete its Cholesky factor. Give a negative direction for [[1,2],[2,1]]. Does replacing positive leading minors by nonnegative ones characterize PSD?

P14. Two error norms. Give the rank-one truncated SVD approximation of diag(4,2,1), its operator error and Frobenius error. For diag(2,2), exhibit two different optimal rank-one approximations and explain why sign equality of factors is not a valid general test.

P15. Center first. Center rows (1,1),(2,2),(3,3). Find singular values of the centered matrix, then sample-covariance eigenvalues using denominator n−1. Compare the raw matrix's largest singular value. Explain the constant-feature case.

After your attempt, use the separate feedback.