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Homework1 - Math Biophysics, Fall 2011 Homework 1 (1) Read...

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Unformatted text preview: Math Biophysics, Fall 2011 Homework 1 (1) Read about finding eigenvalues using Maple. Let −4 −3 3 −3 3 2 −3 3 C= 3 −4 3 3 6 6 −6 5 Use Maple to find a non-zero vector X such that CX = 2X . (2) Use Maple to find the inverses of the matrices 123 2 2−i M = 3 4 5 and N = . 2 + i −2 356 (3) Suppose A is self adjoint and that v1 and v2 are eigenvectors corresponding ∗ to distinct eigenvalues. Show that v1 v2 = 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 A = I and det A = 1, show that for some θ, cos θ − sin θ A= . sin θ cos θ (5) Let Rθ be the rotation matrix cos θ sin θ − sin θ cos θ . Show that (Rθ − 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 λ. (7) Let x(t) = (r cos t, r sin t, p t) be a helix. Show that the curvature is a constant r/(r2 + p2 ) and the torsion is constant p/(r2 + p2 ). (8) Suppose u is a unit vector and B1 and B2 are rotations with B1 e3 = B2 e3 = u. Show that B1 Rz (θ)B−1 = B2 Rz (θ)B−1 . 1 2 (9) Prove that if A is a rotation, then AR (u, θ) A−1 = R (Au, θ) . (10) Show that any rotation A can be written as a product of three rotations about the y and z axes, A = Rz (α)Ry (γ )Rz (δ ). The angles α, γ , δ are called Euler angles for the rotation A. (Hint: Write Ae3 in spherical coordinates.) 1 ...
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This note was uploaded on 01/15/2012 for the course MAP 5485 taught by Professor Staff during the Fall '11 term at FSU.

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