## DESCRIPTION
## Linear Algebra
## ENDDESCRIPTION

## KEYWORDS('Quadratic form' 'Decomposition of Symmetric Matrix')
## Tagged by tda2d

## DBsubject('Linear Algebra')
## DBchapter('Matrices')
## DBsection('Quadratic Forms')
## Date('')
## Author('Shifrin')
## Institution('UGA')
## 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",
"PGauxiliaryFunctions.pl", 
"PGgraphmacros.pl", 
"PGmatrixmacros.pl",
# "PGmorematrixmacros.pl"
"weightedGrader.pl"
);



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

$L = new Matrix(2,2);
$D = new Matrix(2,2);
$LT = new Matrix(2,2);
$A = new Matrix(2,2);

$d11 = non_zero_random(-5,5,1);
$d22 = random(-5,5,1);
$l21 = random(-5,5,1);

$D->assign(1,1,$d11);
$D->assign(2,2,$d22);
$L->assign(2,1,$l21);
$LT->assign(1,2,$l21);

foreach $i (1..2){
foreach $j (1..2){
if($j!=$i) {
$D->assign($i,$j,0);}}}

foreach $i (1..2){
foreach $j (1..2){
if($j==$i) {
$L->assign($i,$j,1); $LT->assign($i,$j,1);} elsif($j>$i){
$L->assign($i,$j,0); $LT->assign($j,$i,0);}}}

$A = $L*$D*$LT;

$ques=("The quadratic form \( \mathcal Q(\mathbf x) = \mathbf x^T A \mathbf x \) is");

if($d11>0 and $d22>0){$ans="positive definite";}
elsif(($d11==0 and $d22>=0) or ($d11>0 and $d22==0)){$ans="positive semidefinite";}
elsif($d11<0 and $d22<0){$ans="negative definite";}
elsif(($d11==0 and $d22 le 0) or ($d11<0 and $d22==0)){$ans="negative semidefinite";}
else {$ans="indefinite";}

$mc0 = new_multiple_choice();
$mc0->qa($ques, $ans);
$mc0->makeLast("positive definite","negative definite","positive semidefinite", "negative semidefinite","indefinite");

BEGIN_TEXT

If \(A=\{display_matrix_mm($A)\}\),
$BR
then we can write \( A = LDL^T \), where
$PAR
\{mbox( '\(L = \)', display_matrix([['\(1\)','\(0\)'],[ans_rule(5),'\(1\)']]) )\}
$BR
\{mbox('and', '\( D = \)', display_matrix([[ans_rule(5),'\(0\)'],['\(0\)',ans_rule(5)]]), '.')\}
$PAR

\{ $mc0->print_q() \}
\{ $mc0->print_a() \}

END_TEXT

WEIGHTED_ANS(num_cmp($l21),25,
num_cmp($d11),25,
num_cmp($d22),25,
radio_cmp($mc0->correct_ans),25);

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

ANS(num_cmp ra_flatten_matrix($C*$B));