NAME

MatrixReduce.pl - reduced row echelon form, row operations, and elementary matrices.

SYNOPSIS

Provides subroutines for elementary matrix computations using MathObjects matrices.

Get the reduced row echelon form: $Areduced = rref($A);

Should be used in the fraction context with all entries of $A made into fractions.

Make matrix entries do fraction arithmetic (rather than decimal arithmetic):

After selecting the Fraction context using Context('Fraction')-parens->set("[" => {formMatrix => 1})>, $A = apply_fraction_to_matrix_entries($A); applies Fraction() to all of the entries of $A, which makes subsequent matrix algebra computations with $A use fraction arithmetic.

Get the reduced column echelon form: $Areduced = rcef($A);

Change the value of a matrix entry: change_matrix_entry($A,[2,3],50);

changes the [2,3] entry to the value 50.

Construct an n x n identity matrix: $E = identity_matrix(5);

(This is an alias for Value::Matrix->I(5);)

Construct an n x n elementary matrix that will permute rows i and j:

$E = elem_matrix_row_switch(5,2,4); creates a 5 x 5 identity matrix and swaps rows 2 and 4.

Construct an n x n elementary matrix that will multiply row i by s:

$E = elem_matrix_row_mult(5,2,4); creates a 5 x 5 identity matrix and swaps puts 4 in the second spot on the diagonal.

Construct an n x n elementary matrix that will multiply row i by s:

$E = elem_matrix_row_mult(5,2,4); creates a 5 x 5 identity matrix and puts 4 in the second spot on the diagonal.

Construct an n x n elementary matrix that will add s times row j to row i:

$E3 = elem_matrix_row_add(5,3,1,35); creates a 5 x 5 identity matrix and puts 35 in the (3,1) position.

Perform the row switch transform that swaps (row i) with (row j):

$Areduced = row_switch($A,2,4); swaps rows 2 and 4 in matrix $A.

Perform the row multiplication transform s * (row i) placed into (row i):

$Areduced = row_mult(A,2,10); multiplies every entry in row 2 of $A by 10.

Perform the row addition transform (row i) + s * (row j) placed into (row i):

$Areduced = row_add($A,2,1,10); adds 10 times row 1 to row 2 and places the result in row 2. (Same as constructing $E to be the identity with 10 placed in entry (2,1), then multiplying $E * $A.)

DESCRIPTION

Usage:

DOCUMENT();
loadMacros(
    "PGstandard.pl",
    "MathObjects.pl",
    "MatrixReduce.pl", # automatically loads contextFraction.pl and MathObjects.pl
    "PGcourse.pl",
);
$showPartialCorrectAnswers = 0;
TEXT(beginproblem());

# Context('Matrix'); # for decimal arithmetic
Context('Fraction'); # for fraction arithmetic

$A = Matrix([
    [random(-5,5,1),random(-5,5,1),random(-5,5,1),3],
    [random(-5,5,1),random(-5,5,1),random(-5,5,1),0.75],
    [random(-5,5,1),random(-5,5,1),random(-5,5,1),9/4],
]);

$A = apply_fraction_to_matrix_entries($A); # try commenting this line out for different results

$Arref = rref($A);

$Aswitch = row_switch($A, 2, 3);

$Amult = row_mult($A, 2, 4);

$Aadd = row_add($A, 2, 1, 10);

$E = elem_matrix_row_add(3,2,1,10);
$EA = $E * $A;

$E1 = elem_matrix_row_switch(5,2,4);
$E2 = elem_matrix_row_mult(5,4,Fraction(1/10));
$E3 = elem_matrix_row_add(5,3,1,35);
$E4 = identity_matrix(4);
change_matrix_entry($E4,[3,2],10);

Context()->texStrings;
BEGIN_TEXT
The original matrix and its row reduced echelon form:
\[ $A \sim $Arref. \]
$BR
The original matrix with rows switched, multiplied, or added together:
\[ $Aswitch, $Amult, $Aadd. \]
$BR
Some elementary matrices.
\[$E1, $E2, $E3, $E4\]
END_TEXT
Context()->normalStrings;

COMMENT('MathObject version.');
ENDDOCUMENT();

AUTHORS

Paul Pearson, Hope College, Department of Mathematics

with help from

Davide Cervone, Union College, Department of Mathematics

Michael Doob, University of Manitoba, Department of Mathematics