Matrixexponential and Normaldistribution/Gauss' function (II)

[Gottfried Helms]

Sorry, I was a bit sloppy with the copy&paste of results.
I have to correct two displays:

Am 04.03.2006 21:54 schrieb Gottfried Helms:
> While the matrix-logarith of the binomial-triangle has the sequence
> of natural numbers n in the *first* subdiagonal, we have here the triangular
> numbers (2n-1)*2n / 2  in the *second* subdiagonal (but only each second
> entry is different from zero.
> 
> I would call things like that maximal compression of information...
> ... and a strong surprise.
> 
> But one can even smooth that thing and proceed;
> use A as the "Gauss-triangle", and compute its square:
> 
>     A1 = A*A
>      	| 1         0         0         0         0         0         0         0        |
>      	| 0         1         0         0         0         0         0         0        |
>      	| -2        0         1         0         0         0         0         0        |
>      	| 0         -6        0         1         0         0         0         0        |
>      	| 12        0         -12       0         1         0         0         0        |
>      	| 0         60        0         -20       0         1         0         0        |
>      	| -120      0         180       0         -30       0         1         0        |
>      	| 0         -840      0         420       0         -42       0         1        |

here I took the wrong output; the numbers must be neagtive;

>  and now the matrix logarithm has no holes in the sequence:
>     lA1 = ln(A*A)
>      	| 0         0         0         0         0         0         0         0        |
>      	| 0         0         0         0         0         0         0         0        |
>      	| 2         0         0         0         0         0         0         0        |
>      	| 0         6         0         0         0         0         0         0        |
>      	| 0         0         12        0         0         0         0         0        |
>      	| 0         0         0         20        0         0         0         0        |
>      	| 0         0         0         0         30        0         0         0        |
>      	| 0         0         0         0         0         42        0         0        |

that means, I forgot to document the inversion-step; it is

       lA1 = ln(inv(A*A))    or lA1 = - ln(A*A)

and as well with
> 
> then the matrix-logarithm has the -say- most simple representation (second only
> to that for the binomial-matrix):
> 
>     LA2 = ln(A2)
should be   LA2 = - ln(A2)   or LA2 = - ln(sqrt(A*A))   or LA2 = ln( inv (sqrt(A*A)))

>      	| 0         0         0         0         0         0         0         0        |
>      	| 0         0         0         0         0         0         0         0        |
>      	| 1         0         0         0         0         0         0         0        |
>      	| 0         3         0         0         0         0         0         0        |
>      	| 0         0         6         0         0         0         0         0        |
>      	| 0         0         0         10        0         0         0         0        |
>      	| 0         0         0         0         15        0         0         0        |
>      	| 0         0         0         0         0         21        0         0        |
> 
and then in the second eq I missed the matrix-inversion, sorry: it must be
> 
>   exp("natural numbers")    = "binomial-triangle"
>                                 _______________________
>   exp("triangular-numbers") = \/inv("Gauß-triangle") ²
which does not affect the numbers, only their sign -
but obviously it loses much of its simplicticy. One can repair that by
>                                   _________________
>   exp(- "triangular-numbers") = \/"Gauß-triangle" ²

it's a pity...


But anyway, I feel the last note is still valid:
> 
> This seems to be the start of an interesting family of numbers/functions...
> 

Gottfried