Re: why does professor david c ullrich have to put people down to feel good about himself?

On Dec 1, 1:55 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
Finally, it's curious that nobody's pointed out
that a "criterion" for establishing that two matrices
commute that depends on two other matrices
commuting seems a little curious.

There are two issues at hand here:

1. Is there a way to establish that A and B commute
that's less computationally expensive than simply
calculating AB and BA?

Based on Ullrich's post here, the answer appears to
be no. Ullrich and others have shown the reasons
that tommy1729's plan is doomed to fail.

First, it was demonstrated that matrix logarithms
are not easy to calculate, much to the dismay of
tommy1729 (and galathaea). The natural method of
power series only works in certain situations.

Second, as Ullrich just posted, why base whether A
and B commute on whether C and D commute? Originally
tommy1729 was hoping that it would be _easier_ to
determine whether C and D commute than whether A and
B commute -- for example, if C happened to commute
with _every_ matrix. It was then pointed out that
very few matrices have that property -- being of the
form lambda I for some lambda -- and so this ended
up being worthless (though the case where lambda is
irrational may be interesting).

In the end, tommy1729's effort turned out to be as
futile as JSH's factoring methods. Notice how JSH's
goal, "surrogate factoring," is to find the factors
of a target in terms of the factors of another
number, the surrogate, just as tommy1729 sought to
determine the commutativity of A and B in terms of
the commutativity of new matrices C and D.

JSH's method failed because it turned out to be
harder to find a surrogate than it was to factor the
number by known methods. And tommy1729's method
failed because it turned out to be harder to find C
and D and establish that they commute than it was
just to find AB and BA.

And now that Israel and Ullrich have revealed the
problems behind tommy1729's methods, let's go on to
the second issue at hand.

2. Is there a way to define exponentiation A^C for
general matrices A and C?

Now this is what I want to discuss. First, tommy1729
mentioned exponentiation. Then galathaea suggested a
definition of A^C as exp(C log(A)), so that A^C has
as many values as A has logarithms. Then Gottfried
Helms suggested that exp(log(A) C) may work better,
especially for infinite matrices.

When I entered this thread, I mentioned A and B
commuting in order to see the motivation behind
tommy1729's mentioning of A^C. But even though A^C
as a means of establishing commutativity has gone by
the wayside, I still think that matrix exponentiation
can be an interesting concept. There may be more to
A^C than just whether A and B commute.

And who knows? Perhaps someday in the distant future,
there may be a math problem whose solution really is
effectively discovered via matrix exponentiation A^C.
.