View of /trunk/NationalProblemLibrary/Rochester/setLinearAlgebra11Eigenvalues/ur_la_11_29.pg

    1 ## DESCRIPTION
    2 ## Linear Algebra
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS ('linear algebra','matrix','eigenvalue','basis','orthogonal')
    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 "PGmorematrixmacros.pl"
   32 );
   33 
   34 TEXT(beginproblem());
   35 $showPartialCorrectAnswers = 1;
   36 
   37 $a= new Matrix(3,3);
   38 
   39 # create an invertible matrix with det either 1  or -1
   40 
   41 $a11 = non_zero_random(-2,2,1);
   42 $a21 = random(-1,1,2);
   43 $a31 = random(-1,1,2);
   44 
   45 $b1 = random(-1,1,2);
   46 $a12 = $b1 * $a11;
   47 $m = random(-1,1,2);
   48 $a22 = $b1 * $a21 + $m;
   49 $a32 = $b1 * $a31;
   50 
   51 $c = random(-1,1,1);
   52 $d = random(-1,1,2);
   53 $n = random(-1,1,2);
   54 $a13 = $c * $a11 + $d * $a12 + $n;
   55 $a23 = $c * $a21 + $d * $a22;
   56 $a33 = $c * $a31 + $d * $a32;
   57 
   58 $det = - $a31 * $m * $n;
   59 
   60 # define matrix
   61 
   62         $a->assign(1,1, $a11 );
   63         $a->assign(1,2, $a12 );
   64         $a->assign(1,3, $a13 );
   65         $a->assign(2,1, $a21 );
   66         $a->assign(2,2, $a22 );
   67         $a->assign(2,3, $a23 );
   68         $a->assign(3,1, $a31 );
   69         $a->assign(3,2, $a32 );
   70         $a->assign(3,3, $a33 );
   71         $a_lr = $a->decompose_LR();
   72         $a_det = $a_lr->det_LR();
   73 
   74 # define inverse matrix
   75         $b = $a_lr->invert_LR();
   76 
   77 # define eigenvalues
   78 
   79         $e = new Matrix(3,3);
   80         $e->one();
   81 
   82 $eig1 = non_zero_random(-4,4,1);
   83 $eig2 = random(-4,4,1);
   84 if ($eig1 == $eig2) { $eig2 = $eig2+1;}
   85   $e->assign(1,1, $eig1);
   86   $e->assign(2,2, $eig1);
   87   $e->assign(3,3, $eig2);
   88 
   89 # define final matrix
   90         $matrix = $a * $e *$b;
   91         $matrix_lr = $matrix->decompose_LR();
   92         $matrix_det = $matrix_lr->det_LR();
   93 
   94 # matrix entries are integers, but we'll round them to avoid printing 0.999999999
   95 
   96 foreach $i (1..3) {
   97   foreach $j (1..3) {
   98     $m[$i][$j] = $matrix->element($i,$j);
   99     $m[$i][$j] = round($m[$i][$j]);
  100     $matrix->assign($i,$j,$m[$i][$j]);
  101   }
  102 }
  103 
  104 BEGIN_TEXT
  105 
  106 \{ mbox( 'The matrix \(A=\)', display_matrix($matrix) ) \}
  107 
  108 $BR
  109 has two real eigenvalues, \(\lambda_1 = $eig1\) of multiplicity \(2\), and \(\lambda_2 = $eig2\) of multiplicity \(1\).
  110 Find an orthonormal basis for the eigenspace \(E_1\).
  111 $BR
  112 \{ mbox( ans_array(3,1,15), ', ', ans_array_extension(3,1,15), '.' ) \}
  113 
  114 END_TEXT
  115 
  116 # a basis of the eigern space
  117 $col1 = $a -> column(1);
  118 $col2 = $a -> column(2);
  119 # now apply Gram-Schmidt'
  120 $scalar = ($col1->scalar_product($col2))/($col1->scalar_product($col1));
  121 $col2 = $col2 - $scalar*$col1;
  122 # now make them unit vectors
  123 $norm1 = $col1->length;
  124 $col1 = (1/$norm1)*$col1;
  125 $norm2 = $col2->length;
  126 $col2 = (1/$norm2)*$col2;
  127 
  128 
  129 ANS(basis_cmp([$col1,$col2], 'mode'=>'orthonormal', 'help'=>'verbose'));
  130 
  131 ENDDOCUMENT();       # This should be the last executable line in the problem.
  132