## DESCRIPTION ## Linear Algebra ## ENDDESCRIPTION ## KEYWORDS ('linear algebra','matrix','eigenvalue') ## Tagged by cmd6a 4/30/06 ## DBsubject('Linear Algebra') ## DBchapter('Matrices') ## DBsection('Eigenvalues') ## Date('') ## Author('') ## 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 = non_zero_random(-2,2,1); \$e = random(1,2,1) + 3*random(-1,1,1) - \$a; \$b = random(-1,1,2); \$sqrD = random(1,2,1) + 3*random(0,2,1); \$D = \$sqrD**2; \$d = (\$D - (\$a+\$e)**2)*\$b/3 + \$a*\$e*\$b; # characteristic polynomial is - (lambda^3 - (a+e)lambda^2 + (ae-bd)lambda - b^2k) # now forget about the minus in front # we want 3 distinct real roots, so we want local max and min with pos and neg values respectively # find max and min: # derivative is 3lamda^2 - 2(a+e)lambda + (ae-bd) # need 2 disctinct real roots # discriminant/4 is (a+e)^2 - 3ae + 3bd = (a+e)^2 - 3ae + D - (a+e)^2 + 3ae = D # ok, the discriminant is positive # roots are \$max_root = (\$a + \$e + \$sqrD)/3; \$min_root = (\$a + \$e - \$sqrD)/3; # local max and min values of the cubic polynomial (without the minus in front) have to be pos and neg, so \$ans_min = \$max_root**3 - (\$a+\$e)*\$max_root**2 + (\$a*\$e - \$b*\$d)*\$max_root; \$ans_max = \$min_root**3 - (\$a+\$e)*\$min_root**2 + (\$a*\$e - \$b*\$d)*\$min_root; BEGIN_TEXT \{ mbox( 'The matrix \(A=\)', display_matrix([[\$a, \$b, 0], [\$d, \$e, \$b], ['k', 0, 0]]) )\} \$BR has three distinct real eigenvalues if and only if \$BR \{ans_rule(20)\} \( < k < \) \{ans_rule(20)\}. END_TEXT ANS(num_cmp(\$ans_min)); ANS(num_cmp(\$ans_max)); ENDDOCUMENT(); # This should be the last executable line in the problem.