= Let p(2) ao+a+ +anz" be an analytic polynomial and let s = 1|2|2 for z C with || 1. Let e, denote the column n x 1 matrix with 1 at the ith place and 0 else. Verify: (1) Consider the (n+1)(n+1) matrix U with columns ze+se2, 3, 4, , en+1, and seize (in order). Then U is unitary with eigenvalues A1, An+1 of modulus 1 (Hint. Check that columns of U are mutually orthonormal). (2) = (e1) Uke (Check: Apply induction on k), and hence p(z) (e1)p(U)e1. (3) max p()|||p(U)|| (Hint. Recall that ||AB|| ||A||||B||) (4) If D is the diagonal matrix with diagonal entries A1, An+1 then ||p(U)|| = ||p(D)||] = max \P(X)|. i=1,...,n+1 Conclude that max||1 |P(2)| = max|2|=1 |P(2)|.