CE 5143 Homework 6 Due 10/25/07

— David Wagner 2007/10/17 11:26

1. Problem #4.33

Prove that the coordinate transformation matrix, given by Eq. (4.50) is orthogonal.

(4.50)delim{[}{P}{]} = delim{[}{ matrix{2}{2}{ {cos theta} {-sin theta} {sin theta} {cos theta} } }{]}

To prove orthogonality of two matrices, show

  1. delim{[}{P}{]}^T delim{[}{P}{]} = delim{[}{I}{]}=
  2. delim{[}{ matrix{2}{2}{ {cos theta} {-sin theta} {sin theta} {cos theta} } }{]}^T delim{[}{ matrix{2}{2}{ {cos theta} {-sin theta} {sin theta} {cos theta} } }{]}=
  3. delim{[}{ matrix{2}{2}{ {cos theta} {sin theta} {-sin theta} {cos theta} } }{]} delim{[}{ matrix{2}{2}{ {cos theta} {-sin theta} {sin theta} {cos theta} } }{]}=
  4. delim{[}{ matrix{2}{2}{ ({cos^2 theta}+{sin^2 theta}) ({sin theta}{cos theta}-{sin theta}{cos theta}) ({-sin theta}{cos theta}+{sin theta}{cos theta}) ({sin^2 theta}+{cos^2 theta}) }}{]}=
  5. delim{[}{ matrix{2}{2}{ 1 0 0 1 }}{]}=delim{[}{I}{]}✔

2. Problem #4.43

Generate the functions fi(λ) for the tridiagonal matrix delim{[}{A}{]} = delim{[}{ matrix{4}{4}{alpha beta 0 0 beta alpha beta 0 0 beta alpha beta 0 0 beta alpha} }{]} with α = 1 and β = 2.

f_0(lambda)=1;

f_1(lambda)=alpha-lambda=1-lambda;

f_2(lambda)=(alpha-lambda)f_1(lambda) -beta^2 f_0(lambda)= (1-lambda)(1-lambda) -2^2(1)= 1-2 lambda +lambda^2 -4= lambda^2 -2 lambda -3;

f_3(lambda)=(alpha-lambda)f_2(lambda) -beta^2 f_1(lambda)= (1-lambda)(lambda^2 -2 lambda -3) -4(1-lambda)= lambda^2 -2 lambda -3 -lambda^3 +2 lambda^2 +3 lambda -4 + 4lambda= -lambda^3 +3 lambda^2 +5 lambda -7;

f_4(lambda)=(alpha-lambda)f_3(lambda) -beta^2 f_2(lambda)= (1-lambda)(- lambda^3 +3 lambda^2 +5 lambda -7) -4 (lambda^2 -2 lambda -3)= -lambda^3 +3 lambda^2 +5 lambda -7 +lambda^4 -3lambda^3 -5lambda^2 +7lambda -4lambda^2 +8lambda +12= lambda^4 -4lambda^3 -6lambda^2 +20lambda +5.

Discussion


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