## DESCRIPTION
## Linear Algebra
## ENDDESCRIPTION

## KEYWORDS ('linear algebra','matrix','symmetric','eigenvalue','eigenvector','orthonormal')
## Tagged by cmd6a 5/3/06

## DBsubject('Linear Algebra')
## DBchapter('Matrices')
## DBsection('Eigenvalues')
## Date('')
## Author('modified by Shifrin')
## Institution('Rochester')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGgraphmacros.pl",
"PGmatrixmacros.pl", 
"PGnumericalmacros.pl",
"PGauxiliaryFunctions.pl",
"PGmorematrixmacros.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,9,1);
$b = random(-9,-1,1);

BEGIN_TEXT

The matrix \( A =  \{display_matrix_mm([[0, 0, $a], [0, $b, 0], [$a, 0, 0]]) \} \) 
$PAR
has two distinct eigenvalues \(\lambda_1 < \lambda_2\). 
Find the eigenvalues and an orthonormal basis for each eigenspace.
$BR
\{ mbox( '\(\lambda_1 \) = ', ans_rule(10) , ',' ) \} 
$BR
\{mbox( 'Orthonormal basis: ', ans_array(3,1,10), ',' ) \}
$BR	
\{ mbox( '\(\lambda_2 \) = ', ans_rule(10), ',' ) \}
$BR
\{mbox( 'Orthonormal basis: ', ans_array(3,1,10), ',', ans_array_extension(3,1,10), '.' ) \}
$BR
The above eigenvectors form an orthonormal basis for \(\mathbb R^3\). 

END_TEXT

ANS(num_cmp($b));
ANS(basis_cmp([[0, 1, 0]], 'mode'=>'unit', 'help'=>'verbose'));
ANS(num_cmp($a));
ANS(basis_cmp([[1, 0, 0], [0, 0, 1]], 'mode'=>'orthonormal', 'help'=>'verbose'));

ENDDOCUMENT();       # This should be the last executable line in the problem.