The nonzero singular values of a matrix A ∈ R10×10 are 1 12 13 · · · 19 a What is the spectral norm of Ab Let x ∈ R10 be a unit vector and let k ∈ 1 · · · 9 be given What values of k wouldguarante

(a) The spectral norm of A is the maximum singular value, which is 1.9.

(b) We have ∥Ax - Akx∥ = ∥A(I - Ak)x∥ ≤ ∥A∥ ∥(I - Ak)x∥, where ∥A∥ is the spectral norm of A. Since x is a unit vector, we have ∥(I - Ak)x∥ = |1 - ak|, where ak is the kth diagonal entry of A. Therefore, we need |1 - ak| ≤ 0.2/∥A∥ = 0.2/1.9. Solving for k, we get 4 ≤ k ≤ 6.