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Homework1

# Homework1 - Math Biophysics Fall 2011 Homework 1(1 Read...

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Math Biophysics, Fall 2011 Homework 1 (1) Read about finding eigenvalues using Maple. Let C = - 4 - 3 3 - 3 3 2 - 3 3 3 3 - 4 3 6 6 - 6 5 Use Maple to find a non-zero vector X such that CX = 2 X . (2) Use Maple to find the inverses of the matrices M = 1 2 3 3 4 5 3 5 6 and N = 2 2 - i 2 + i - 2 . (3) Suppose A is self adjoint and that v 1 and v 2 are eigenvectors corresponding to distinct eigenvalues. Show that v * 1 v 2 = 0. (First read the proof in the Brief Linear Algebra Review that eigenvalues of selfadjoint matrices are real. Use a similar method of proof.) (4) If A is a real 2 × 2 matrix such that A 0 A = I and det A = 1, show that for some θ , A = cos θ - sin θ sin θ cos θ . (5) Let R θ be the rotation matrix cos θ - sin θ sin θ cos θ . Show that ( R θ - I ) ( R - θ - I ) = 2(1 - cos θ ) I. (6) Show that if A is real and has real eigenvalue λ , then there is a real vector (a vector with real coordinates) which is an eigenvector corresponding to
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