MMAT5320 Computational Mathematics Take Home Final Exam • There are a total of 3 questions, and the total score is 126 points. • This is to make up from the abrupt end of the semester. There will be no questions on the material in the last chapter on Eigenvalue algorithms. • Please write down your answers in pen (no pencil work will be accepted), then provide a scan of your answers and email them to me at [email protected] • You must send your answers via email by 20th December 2019 23:59pm. • Late answers will not be accepted! There will be no exceptions. Question 1 (a) [8 pts] Let A ∈ Rm×n be a matrix of full rank. Write down the procedure to obtain a reduced SVD of A. (b) [10 pts] Compute a reduced SVD of the following matrix A = ⎛⎜⎜⎜⎝ 2 0 1 2 0 1 0 0 ⎞⎟⎟⎟⎠ . (c) [8 pts] Find two more vectors u⃗3 and u⃗4 in R4 such that the set {u⃗1, . . . , u⃗4} with the following u⃗1 and u⃗2 forms an orthonormal set: u⃗1 = 1√ 14 ⎛⎜⎜⎜⎝ 2 3 1 0 ⎞⎟⎟⎟⎠ , u⃗2 = 1√ 6 ⎛⎜⎜⎜⎝ 2−1−1 0 ⎞⎟⎟⎟⎠ . In your answer demonstrate clearly that {u⃗1, . . . , u⃗4} satisfies all the requirements to be an orthonormal set. (d) [3 pts] Write down a full SVD of the matrix A in part (b). Question 2 (a) [20 pts] Let {a⃗1, . . . , a⃗n}, where a⃗i ∈ Rm for 1 ≤ i ≤ n and m > n, be a set of linearly independent vectors. Describe one method to derive a full QR factorisation of the matrix A whose columns are {a⃗1, . . . , a⃗n}. You may choose from the standard Gram– Schmidt orthonormalisation procedure, Gram–Schmidt projections or Householder reflections. 1 In your answer, outline the details for step 1, step 2 and for step k, and show how to construct the matrices Q and R. (b) [13 pts] Compute the reduced QR factorisation of the following matrix B = ⎛⎜⎜⎜⎝ 1 5 2 1 −1 2 1 5 4 1 −1 4 ⎞⎟⎟⎟⎠ . You may use any procedure for your calculations (Gram–Schmidt, Gram–Schmidt projections, Householder reflections). (c) [6 pts] Compute the pseudoinverse corresponding to B, and compute the least squares solution x⃗∗ ∈ R3 to the problem ∥Bx⃗∗ − b⃗∥2 ≤ ∥By⃗ − b⃗∥2 for any y⃗ ∈ R3 if b⃗ = (1,1,−1,−1)⊺. (d) [9 pts] Let x⃗1 and x⃗2 be two linearly independent vectors in R3, and let P = span{x⃗1, x⃗2} denote the plane spanned by these two vectors. Let X ∈ R3×2 be the matrix whose columns are x⃗1 and x⃗2, and let X = QR be a full QR factorisation. • Show that the first two columns of Q, denoted by q⃗1 and q⃗2, lie in the plane P . • Use the full QR factorisation to find a normal direction to the plane P . Question 3 (a) [20 pts] • Given an example of a 3 × 3 matrix which is not diagonalisable. • Given an example of a 2 × 2 matrix which is diagonalisable, but not unitary diagonalisable. In your answer you must demonstrate why your chosen matrix satisfies the required properties. (b) [20 pts] Compute the eigenvalues and the Schur factorisation of the following matrix A = ⎛⎜⎜⎜⎝ 2 0 0 0−1 3 0 0 0 0 6 7 0 0 0 8 ⎞⎟⎟⎟⎠ (c) [9 pts] For a general square matrix A ∈ Rm×m, let A = QTQ⊺ denote a Schur factori- sation with orthogonal Q and upper triangular T , where q⃗i, 1 ≤ i ≤ m denotes the ith column of Q. Show that the second column q⃗2 is an eigenvector to the matrix B ∶= (I − q⃗1q⃗⊺1)A(I − q⃗1q⃗⊺1) and compute the corresponding eigenvalue. 2
