Primes approximated by eigenvalues?
Solution 1:
This is not an answer, but too long for a comment.
Are you aware, that the L-D-U-decomposition gives triangular matrices, whose entries are constant for any dimension? That means the matrices of finite sizes can be seen as simply truncated versions of the infinite matrix, and for the matrices with size going to infinity should then approximate the primes by its eigenvalues arbitrarily exact.
Also because T is symmetric, the L and U-factors are transposed of each other.
Here is the top lft of the L-matrix (the left column is the row-index):
$$ \small \small \begin{array} {rr}
\begin{array} {r} 1 \\ 2 \\3\\4 \\5 \\6 \\7 \\8 \\9 \\10 \\11 \\12 \\ \end{array} &
\begin{bmatrix}
1 & . & . & . & . & . & . & . & . & . & . & . \\
1 & 1 & . & . & . & . & . & . & . & . & . & . \\
1 & 0 & 1 & . & . & . & . & . & . & . & . & . \\
1 & 1 & 0 & 1 & . & . & . & . & . & . & . & . \\
1 & 0 & 0 & 0 & 1 & . & . & . & . & . & . & . \\
1 & 1 & 1 & 0 & 0 & 1 & . & . & . & . & . & . \\
1 & 0 & 0 & 0 & 0 & 0 & 1 & . & . & . & . & . \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & . & . & . & . \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & . & . & . \\
1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & . & . \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & . \\
1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix} \end{array}$$
and the diagonal-matrix D $$ \small \small \operatorname{diag} (D)=\begin{bmatrix} -1 & 2 & 3 & 0 & 5 & -6 & 7 & 0 & 0 & -10 & 11 & 0 \end{bmatrix} $$
It is nice to see the rows in L at the prime row-indexes, in general they seem to be indicators of the primefactors of the row-index and the D is not perfectly clear to me: the sign giving the number of different primefactors?). If we had the true patterns then this would allow to generate the L and the D matrices directly; the construction of the T by the recursive procedure is much time-consuming!
[update] With a bit improved heuristic the pattern of the entries in D seems to be
$ \qquad \qquad \small D_1 = -1 $
$ \qquad \qquad \small D_k = 0 \qquad \text{if k is not squarefree}$
$ \qquad \qquad \small D_k = -k \dot (-1)^{w(k)} \qquad \text{if k is squarefree}$
where $w(k)$ is the number of distinct primefactors. Using the Moebius-function we can write $$ D_k = -k \cdot \operatorname{Moebius}(k) $$
The entries in L seem to be: for a row r at a column c we have 1 if $c=r$ or if (c is squarefree and is a divisor of r).
A routine in Pari/GP without recursive call which produces quickly the matrix of size mxm:
{make_T(m)=local(M);M=matrix(m,m);
for(c=1,m,M[1,c]=M[c,1]=-1);
for(r=2,m, for(c=2,r, M[r,c]=M[c,r]= - sum(i=1,c-1,M[r-i,c])));
return(M);}