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Fiddling with another (older) question here I constructed an example-matrix of the type
$\small M_n: m_{n:r,c} = {1 \over (1+r)^c } \quad \text{ for } r,c=0 \ldots n-1 $ . I considered the inverse W : $\small W_n=M_n^{-1} $ for some small n and observed, that the entries in W from the left columns on tend to zero. Using the LDU-decomposition and inverting that L,D,U-factors, one can observe a simple pattern for the coefficients in the matrices or better: for the terms of the dot-products of that inverses in the leftmost columns which is independent of the matrixsize.

For instance, if we denote the inverted L,D,U-factors as K, C, T such that for any n $\small W_n = M_n^{-1} = T \cdot C \cdot K = U^{-1} \cdot D^{-1} \cdot L^{-1}$ then the top-left entry in $\small W_n $ can be computed by the dot-product of the first row of T , C and the first column of K which shows a simple pattern such that we assume the following type of sum: $$\small w_{n:0,0} = \sum_{j=0}^{n-1} (-1)^j (j+1)/j! \qquad \text{ and } \qquad \lim_{n\to \infty} w_{n:00}=0 $$ Now for the two first columns it seems, that indeed all dot-products vanish when n increases and that this can be shown by relatively simple modification and linear composition of the formal exponential-series. But for the next columns this becomes more difficult because the patterns are complicated (but seem to be recursive) and the matrices need increasing size to actually approximate the limiting values.

So the question:a) does W indeed approach the Null-matrix as n goes to infinity?
or in more detail: what are the patterns for the dot-products which occur in the evaluation of the single entries in W ?


I seem to have solved the pattern for the matrix of the partial product $\small C \cdot K $ by a good heuristic. Let's denote the entries with the small letter g then

$$ g_{r,c} = (-1)^c \cdot (1+c)^r \binom{r+1}{c+1} $$

Because at least the first two rows in T are simple to decode $$ \begin{eqnarray} t_{0,c} &=& (-1)^{r+c} {1 \over c!} \\ t_{1,c} &=& (-1)^{r+c} \binom{c+1}{2}{1 \over c!} \end{eqnarray} $$ the results by the two dot-products for rows r=0 and r=1
$$ \begin{eqnarray} w_{0,c} = \sum_{k=0}^{\infty} t_{0,k} \cdot g_{k,c} = 0 \\ w_{1,c} = \sum_{k=1}^{\infty} t_{1,k} \cdot g_{k,c} = 0 \\ \end{eqnarray} $$

give for the first two rows in W zeros for the first couple of columns (with index c=0..4 checked using wolframalpha and the formulae)

Wolframalpha could even give answers for the general column (colindex=c kept indeterminate) For the rows r=0 and r=1 in W for all columns c the value $\small w_{r,c}=0 $
Here are the Wolframalpha formulae:
$$ \sum_{k=c}^{\infty} \left[ \left({(-1)^k \over k!}\right) * \left((-1)^ c (1 + c )^k \binom{k+1}{ c +1}\right) \right] $$ (input) $$ \sum_{k=c}^{\infty} \left[ \left({(-1)^k \over k! }\binom{k+1}{2} \right) * \left((-1)^ c (1 + c )^k \binom{k+1}{c +1} \right) \right] $$ (input)

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[Update]: I've updated the investigations in the *.pdf -file which was already linked below

I've now found a very good heuristic for an analytic description of the inverses of the LDU-decomposition and thus a description for each $\small w_{r,c}$ - element of the sought matrix $\small W=M^{-1}$ . That descriptions are infinite sums involving Stirling numbers first kind, binomials and factorials only. For the two first rows Mathematica at wolframalpha could solve that sums analytically to be zero for the complete rows (keeping the columnindex c indeterminate). For the other entries the symbolic description is possibly too complex, but each element $\small w_{r,c} $ which I tested separately via Mathematica came out to be analytically zero. For a more detailed answer see the exposition of the problem and of the solution in Is the inverse the Null-matrix?

The question has now a good heuristical partial answer, but still a complete answer is not yet in sight. Perhaps the found result is also a new (and infinite) set of identities for Stirling-numbers first kind...

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