Math 104, Summer 2010 Homework 2 1. (Trefethen–Bau 1.3). Suppose a matrix R with entries rij is upper-triangular (i.e., rij = 0 for all i > j). Suppose further that R is m × m and invertible. By using equation 1.8 on the top of p. 8 of the text and considering the span of the first j columns of R, show that R−1 is also upper-triangular. The same result is also true if we replace “upper-triangular” by “lower-triangular” (rij = 0 for i < j) but you only need to prove it in the upper-triangular case. 2. Let u, v be vectors in Rn such that ||u|| = 3, ||v|| = 4. What are the smallest and largest values of u · v? 3. (a) Is the matrix 1 1 −1 A= 2 1 −1 1 −1 1 1 1 1 1 1 −1 1 −1 −1 unitary? 1 0 (b) Write b = 0 as a linear combination of the columns of A. 0 (c) Find the orthogonal projection of b onto span{a1 , a2 , a3 }, where a1 , ..., a4 denote the columns of A. 4. (Trefethen–Bau 2.1) Show that if a matrix A ∈ Cm×m is triangular (i.e., upper-triangular or lower-triangular, as defined in problem 1) and unitary, then it is diagonal (i.e., its entries aij are 0 for all i 6= j). 5. (Trefethen–Bau 2.3) We say a matrix A ∈ Cm×m is hermitian if A∗ = A. Recall that λ ∈ C is an eigenvalue of A if Ax = λx for some x 6= 0. (a) Prove that if A is hermitian, then the eigenvalues of A must be real. (Hint: consider x∗ Ax.) (b) Prove that if A is hermitian and x, y are eigenvectors corresponding to distinct eigenvalues of A, then x and y are orthogonal. (Hints: consider x∗ Ay; you may also want to use part (a).) 6. (Trefethen–Bau 2.4) What can you say about the eigenvalues of a unitary matrix? 7. Prove that (AT )−1 = (A−1 )T . 1