Again consider the same infinite matrix above starting:
T(n,k) = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\ +1&+1&+1&+1&+1&+1&-6 \end{bmatrix}
It then appears that the...
I don't know of any matrix that has all the primes as its only eigenvalues. But there appears to be a matrix such that its most negative eigenvalue (one eigenvalue per matrix) is a prime plus minus a small number.
T(n,1)=1, T(1,k)=1, n>=k: -\sum\limits_{i=1}^{k-1} T(n-i,k), n<k...