## 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.