fuzzy test for row equivalency

[Alex-Jordan]

This adds an isREQ method for Matrix objects. Used like $A->isREQ($B).

This is a "fuzzy " check that avoids both machine rounding issues and student rounding issues (for example if they enter something like 0.3333 instead of 1/3). Instead of reducing each matrix to RREF (which could introduce extraneous pivot positions from machine rounding), this approach compares the two row spaces. If each row from one matrix is "close enough" to the other matrix's row space, then the algorithm effectively declares that row to be in the other matrix's row space. "Close enough" is based on normalizing nonzero rows to length 1, and then the usual fuzzy tolerances for comparing that normalized row and its projection onto the other matrix's row space.

For example, Matrix([ 3, 1 ], [ 1, 1 / 3 ]) and Matrix([ 1, 0.3333 ], [ 0, 0 ]) will be considered row equivalent.

But Matrix([ 3, 1 ], [ 1, 1 / 3 ]) and Matrix([ 1, 0.33 ], [ 0, 0 ]) will not be.

And of course Matrix([ 3, 1 ], [ 1, 1 / 3 ]) and Matrix([ 1, 0], [ 0, 1 ]) will not be.

For questions that ask a student to completely reduce $A, a checker could check $student->isRREF() && $student->isREQ($A).

[Alex-Jordan]

I updated the top comment here, partly to fix typos, but also to explain a little more about what the comparison actually does.

In another thread, @somiaj and I were talking about actually implementing an RREF algorithm. We could do that too, I just couldn't think of a way to avoid extraneous pivot positions popping up from machine rounding. That situation would create "false negatives" where two matrices that should be considered row equivalent would not be. But here, the algorithm goes the other way. It may create "false positives" because of the fuzzy tolerances.

[drgrice1]

I might be misunderstanding something, but I tested with the following problem:

DOCUMENT();

loadMacros('PGstandard.pl', 'MathObjects.pl', 'PGML.pl');

$A = Matrix([ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]);
$B = Matrix([ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ], [ 14, 16, 18 ] ]);

BEGIN_PGML
[``[$A]``] and [``[$B]``] are [$A->isREQ($B) ? 'row equivalent' : 'not row equivalent']
END_PGML

ENDDOCUMENT();

and I get the error Error evaluating variable $A->isREQ($B) ? 'row equivalent' : 'not row equivalent': Cannot compare row equivalency because matrices differ in the first dimension. It seems that these matrices should be comparable, and should be row equivalent. They have the same row space.

[Alex-Jordan]

I could change it for that interpretation. But the implementation is based on A~B meaning that you could apply row operations to one to get the other. And applying row operations to a 3x3 matrix can't produce a 4x3 matrix. Even if they have the same row space.

[drgrice1]

I see. Yeah, that is the standard definition of row equivalent matrices.