View of /trunk/NationalProblemLibrary/Rochester/setLinearAlgebra11Eigenvalues/ur_la_11_11.pg

    1 ## DESCRIPTION
    2 ## Linear Algebra
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS ('linear algebra','matrix','eigenvalue','eigenvector')
    6 ## Tagged by cmd6a 4/30/06
    7 
    8 ## DBsubject('Linear Algebra')
    9 ## DBchapter('Matrices')
   10 ## DBsection('Eigenvalues')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGchoicemacros.pl",
   26 "PGanswermacros.pl",
   27 "PGgraphmacros.pl",
   28 "PGmatrixmacros.pl",
   29 "PGnumericalmacros.pl",
   30 "PGauxiliaryFunctions.pl"
   31 );
   32 
   33 TEXT(beginproblem());
   34 $showPartialCorrectAnswers = 1;
   35 
   36 $u1 = random(-5,5,1);
   37 $v1 = random(-5,5,1);
   38 $u2 = random(-5,5,1);
   39 $v2 = random(-5,5,1);
   40 
   41 $det = $u1*$v2 - $v1*$u2;
   42 
   43 while ($det == 0) {
   44   $u1=$u1+1;
   45   $v2=$v2+1;
   46   $det = $u1*$v2 - $v1*$u2;
   47 }
   48 
   49 $e = random(1,10,1);
   50 $f = random(-10,-1,1);
   51 
   52 ## matrix [u v] is invertible, Au=eu, Av=fv, so A[u v] = [eu, fv]
   53 ## A = [eu, fv] [u v]^{-1}
   54 
   55 $a11 = ($e*$u1*$v2 - $f*$v1*$u2)/$det;
   56 $a12 = (- $e*$u1*$v1 + $f*$v1*$u1)/$det;
   57 $a21 = ($e*$u2*$v2 - $f*$v2*$u2)/$det;
   58 $a22 = (- $e*$u2*$v1 + $f*$v2*$u1)/$det;
   59 
   60 BEGIN_TEXT
   61 
   62 Find a \(2\times 2\) matrix \(A\) such that
   63 $BR
   64 \{ mbox( display_matrix([[$u1], [$u2]]), ' and ', display_matrix([[$v1], [$v2]]) ) \}
   65 $BR
   66 are eigenvectors of \(A\), with eigenvalues \($e\) and \($f\) respectively.
   67 $BR
   68 \{ mbox( '\(A=\)', answer_matrix(2,2,15), '.' ) \}
   69 
   70 END_TEXT
   71 
   72 ANS(num_cmp($a11));
   73 ANS(num_cmp($a12));
   74 ANS(num_cmp($a21));
   75 ANS(num_cmp($a22));
   76 
   77 ENDDOCUMENT();       # This should be the last executable line in the problem.
   78