Matrix Operator Method

[Gottfried]

Thanks for this repetition course, Gottfried, now I first time understand what you are talking about all the time I mean I still knew what an Eigenvector was from my base lessons, but it was never clear to me what you mean by Eigenanalysis etc, now this is clear.

Well, thanks, so it was useful, although my terminology is often a bit handwaved... :-). Good!

Did you anyway already realize that instead of
\( \exp(t\cdot\log(A)) \) you can directly use the binomial series for computation?
\( A^t = \sum_{n=0}^\infty \left(t\\n\right) (A-I)^n \)

No, but it looks very good. I'll give it a deeper look, thanks for the hint!

[update]
Well, in a second thought: in the form of
\( (A+I)^t = \sum_{n=0}^\infty \left(t\\n\right) A^n \)
actually I used it extensely for the scalar and also for the vectorial case.
Applied to a vector of type V(x) = [1,x,x^2,...] it is
P * V(x) = V(x+1)
where P is the pascal-matrix containing the binomial-coefficients.
I dealt a lot with this, and analoguously I derived the fractional and complex powers of P for the general complex solution
P^s * V(x) = V(x + s)
via matrix-logarithm of P (which is an extremely basic object, btw! see this on a t-shirt :     and in wikipedia at "pascal-triangle" )
Then, with iterated application of P as an operator, just equivalently as described here related to tetration, one can find the interesting Faulhaber/Bernoulli-matrix and zeta/eta-values at negative exponents.
Summing of like powers which may be seen as a preliminary training :-)

Gottfried