Re: tetration, matrix^matrix, log(matrix)*matrix or matrix*log(matrix)

Gottfried wrote :

Am 25.12.2008 22:00 schrieb Gottfried Helms:
then there is a non-invertible matrix W where

P~ * W = W * E // E is
diag(1,e,e^2,e^3,...)

and W is the vandermonde-matrix
W = matrix(r=0..inf,c=0..inf c^r/r! )
//r=rowindex,c=ol-index

It should be noted as a curiosity, that P is
a triangular matrix with unit-diagonal. For finite
dimension the eigenvalues for such a matrix are
uniquely determined and they equal the entries of the
matrix-diagonal.
In the case of infinite dimension, using a somehow
"generalized" eigenvector-matrix, it seems, we can
have eigenvalues, which differ from the triangular
matrix-diagonal. So we have a nice example here for
a new property of infinite matrices compared to
finite matrices.

This is even more interesting, since we had from the
diagonalization-approach to tetration, that we
expected
*different* (infinite) sets of eigenvalues, which
reflect
the different possible fixpoint-shifts. From the
consideration
of finite matrices we could not even think about
different sets
of eigenvalues...

Hmmm...

Gottfried Helms

hmm

is there a name for this property ?

or is this what you call fixpoint shift ?

i cant find much about infinite matrices ...

the concept only seems to occur in connection to tetration ?

intresting.

perhaps post it to tetration forum.

and give me some credit , since i came with A^C :)


marry Xmas

regards

tommy1729
.

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