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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 nonzero 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.
 Fall '11
 staff

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