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

David C. Ullrich a écrit :
On Sun, 30 Nov 2008 22:35:28 -0800 (PST), lwalke3@xxxxxxxxx wrote:

On Nov 30, 2:23 am, Denis Feldmann <denis.feldmann.sanss...@xxxxxxx>
wrote:
lwal...@xxxxxxxxx a écrit :
By now, I'm starting to wonder whether there's _anything_ that
Ullrich will accept as a 100% rigorous, 100% valid definition
of A^C.
For instance, one could start by =reading= about log of matrices, or
even before that, about eigenvalues...
I already know about eigenvalues. I already know that
for a matrix A, if there exists a scalar lambda and a
nonzero vector v such that Av = lambda v, then we
call lambda an eigenvalue (my old linear algebra
teacher abbreviated this "eit" for some strange
reason") and v an eigenvector of the matrix. And if
there are as many linearly independent vectors as the
dimension of the matrix, then we can diagonalize the
matrix A = PDP^-1, where P is a matrix whose column
vectors are eigenvectors of A and D is a diagonal
matrix with the eits of A on the diagonal.

Does tommy1729 know about eigenvalues? I don't know,
but as mentioned in the other thread, many of the
properties of the matrix necessary for finding a
logarithm, that tommy1729 ascribes to the determinant
of the matrix, are actually more applicable to the
eigenvalues of the matrix.

Feldmann assumed that I knew nothing about eigenvalues
because my posts are based on tommy1729's, and he
fails to mention them.

Maybe we should even change the conjecture so that it
refers to eigenvalues, not determinants:

For all matrices A,B with _eigenvalues_ not 0,+/-1,
AB = BA iff there exist matrices C,D such that
exp(D) = A, CD = DC, and exp(CD) = B.

As long as Robert's already given everything away,
someone should point out two things: First, the
condition about eigenvalues not equalling 1 or -1
is not needed - I've asked a few times and nobody's
explained why anyone would imagine that that
had any relevance.



Easy : the numbers 1 and -1 appears at one or two places here (like at bondaries for convergence of the series for ln (1+x). For people not knowing (and not willing to do) any math, and juist looking (not even very deeply) at Web references, it seems like a reasonable condition to add "just in case"...

Second, it's easy to see that the conjecture is
equivalent to a more symmetric version: Given
invertible matrices A and B, we have AB = BA
if and only if there exist C and D with CD = DC
and A = exp(C), B = exp(D).

(Note I didn't say that the existence of C and D as in
your version is equivalent to the existence of C and
D as in my version.)

Again, this is easy : your version looks correct. Otoh, what do you do for non invertible matrices? Got you, there...


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.

Curiouser and curiouser. But then, the usual criterions of tommy (or si it Timmy, or amy? we need a criterion here too : do those aliases commute?) are much more convoluted : remember the one he gave for analicity a while ago?


David C. Ullrich

"Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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