This article is meant for readers who, like me, have studied Linear Algebra and who, like me, are curious about Quantum Mechanics.
Are the columns of matrices in a given, natural order, as we write them? Well, if we are using the matrix as a rotation matrix in CGI – i.e. its elements are derived from the trig functions of Euler Angles – then the column order depends, on the order in which we have labeled coordinates to be X, Y and Z. We are not free to change this order in the middle of our calculations, but if we decide that X, Y and Z are supposed to form a different set, then we need to use different matrices as well.
(Edited 02/15/2018 :
OTOH, We also know that a matrix can be an expression of a system of simultaneous equations, which can be solved manually through Gauss-Jordan Elimination on the matrix. If we have found that our system has infinitely many solutions, then we are inclined to say that certain variables are the “Leading Variables” while the others are the “Free Variables”.
It is being taught today, that the Free Variables can also be made our parameters, so that the set of values for the Leading Variables follows from those parameters. But wait. Should it not be arbitrary for certain combinations of variables, which follows from which?
The answer is, that if we simply use Gauss-Jordan Elimination, and if two variables are connected as having possibly infinite combinations of values, then
it will always be the variables stated earlier in the equations which become the Leading, and the ones stated later in the equations will become the Free Variables. We could restate the entire equations with the variables in some other order, and then surely enough, the variable that used to be a Free one will have become a new Leading one, and vice-versa. (And if we do so, the parametric equations for the other Leading variables will generally also change.)
As of 02/15/2018:
This is an observation which I once made, based on certain exercises in Linear Algebra, as taught, having been simplified in the way I described. Eventually, systems of solutions will come up in the real world, in which a Free Variable actually precedes a Leading Variable, both in the order they get mentioned in equations, as well as according to the order of matrix-columns. The corresponding row with a single 1, corresponding to those Free Variables, will not occur, so that it also cannot be added to or subtracted from what will be the earlier row, in such a case.
The later observation follows from the fact that such solutions have infinitely many solutions:
- If the Free Variables are just given a set of values that follow from the solution-set, Then the Leading variables would still need to have definite values, as defined by the same solution-matrix,
- If the Free Variables are given a set of values, that no longer follow from the solution-set, it will not follow that a different set of values for the Leading Variables, will make such extraneous solutions viable.
End of edit, 02/15/2018 )
The order of the columns, has become the order of discovery.
This could also have ramifications for Quantum Mechanics, where matrices are sometimes used. QM used matrices at first, in an effort to be empirical, and to acknowledge that we as Humans, can only observe a subset of the properties which particles may have. And then what happens in QM, is that some of the matrices used are computed to have Eigenvalues, and if those turn out to be real numbers, they are also thought to correspond to observable properties of particles, while complex Eigenvalues are stated – modestly enough – not to correspond to observable properties of the particle.
Even though this system seems straightforward, it is not foolproof. A Magnetic North Pole corresponds according to Classical Principles, to an angle from which an assumed current is always flowing arbitrarily, clockwise or counter-clockwise. It should follow then, that from a different perspective, a current which was flowing clockwise before, should always be flowing counter-clockwise. And yet according to QM, monopoles should exist.
Continue reading Whether the Columns of Matrices have a Natural Order