Linear transformations that retain two by two positive definiteness.
Posted by peeterjoot on October 4, 2009
Purely for fun, lets study the classes of linear transformations that retain the positive definiteness of a diagonal two by two quadratic form. Namely, the Hamiltonian
under a change of variables that mixes position and momenta coordinates in phase space
We want the conditions on the matrix such that the quadratic form retains the diagonal nature
which in matrix form is
So the task is to solve for the constaints on the matrix elements for
Strictly speaking we can also scale and retain positive definiteness, but that case is not of interest to me right now so I’ll use this term as described above.
The expectation is that this will neccessarily include all rotations. Will there be any other allowable linear transformations? Written out in full we want the solutions of
Written out explicitly we have three distinct equations to reduce
Solving for in 9 we have
So, provided , we have a first simplifying identity
Written out to check, this reduces our system of equations
so our equations are now
There are two cases to distinguish here. The first is the more trivial case, for which we find . For the other case we have
Again, writing out in full to check, this reduces our system of equations
We have now only one constraint left, and have reduced things to a single degree of freedom
We’ve already used to distinguish the roots of , so here lets imply that this square root can take either positive or negative values, but that we are treating the sign of this the same whereever seen. Our transformation, employing as the free variable is now known to take any of the following forms
The last of these (the case from earlier) was not considered, but doing so one finds that it produces nothing different from the second form of the transformation above. That leaves us with two possible forms of linear transformations that are allowable for the desired constraints, the first of which screams for a trigonometric parameterization.
For we can parameterize with . Should we allow complex valued linear transformations? If so is a reasonable way to parameterize the matrix for the case. The complete set of allowable linear transformations in matrix form are now
There are really four different matrixes in each of the above. Removing all the shorthand for clarity we have finally
The last four possibilities are now seen to be redundant since they can be incorporated into the cases of the real trig parameterizations where , and . Employing a change of variables, we find that two of the hyperbolic parameterizations are also redundant and can express the reduced solution set as
I suspect this class of transformations has a name in the grand group classification scheme, but I don’t know what it is.