MET Workshop: Exercises

• Rd is endowed with the standard inner product (·, ·) and Euclidean norm ‖ · ‖. • Md×d(R) denotes the space of n× n real matrices. • When Rd = E ⊕ F is a splitting of Rd into complementary subspaces, we let πE//F denote the projection onto E, parallel to F ; that is, the range of πE//F is E, πE//F ◦ πE//F = πE//F , and kerπE//F = F . • For a subspace E ⊂ Rd, let πE denote the orthogonal projection onto E. • For a subspace E ⊂ Rd, let E⊥ denote the orthogonal complement to E. • For 1 ≤ k ≤ d and a linear operator A ∈ Md×d(R), let σk(A) denote the k-th singular value of A.