MATH36001.pdf - MATH36001 Two Hours THE UNIVERSITY OF...

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MATH36001 Two Hours THE UNIVERSITY OF MANCHESTER MATRIX ANALYSIS 25 January 2017 14:00 – 16:00 Answer ALL five questions in Section A (40 marks in all) and TWO questions in Section B (20 marks each). The total number of marks on the paper is 80. If more than TWO questions from Section B are attempted, credit will be given for the FIRST TWO answers. Electronic calculators are not permitted. 1 of 5 P.T.O.
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MATH36001 SECTION A Answer ALL five questions A1. Let A C n × n . (a) State Schur’s theorem for A . (b) Recall that A is normal if AA * = A * A . Show that if A admits the decomposition A = UΛU * with U unitary and Λ diagonal then A is normal. [5 marks] A2. Let u, v be nonzero vectors in C n with n > 1. (a) Determine rank( uv * ) and det( uv * ). (b) Verify that k uv * k F = k u k 2 k v k 2 , where k·k F denotes the Frobenius norm and k·k 2 is the vector 2-norm. (c) Determine the eigenvalues of I + uv * , where I is the n × n identity matrix. Construct the characteristic polynomial and the minimal polynomial of uv * . [10 marks] A3. Let J = λI + N be any 3 × 3 Jordan block, where N = 0 1 0 0 0 1 0 0 0 and I is the 3 × 3 identity matrix. Show that e N = I + N + N 2 2 and e J = e λ e λ e λ / 2 0 e λ e λ 0 0 e λ . [5 marks] A4. Let A C m × n and b C m be given. The pseudoinverse of A , denoted by A + , is the unique n × m matrix satisfying the four Moore-Penrose conditions: (i) AA + A = A, (ii) A + AA + = A + , (iii) AA + = ( AA + ) * , (iv) A + A = ( A + A ) * .
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