I'm curious about the matrix mentioned in a now-closed Stackoverflow question. The matrix has the form:
1, 0, 0, 0, ... 0 2, 2, 0, 0, ... 0 3, 3, 3, 0, ... 0 ... n - 1, n - 1, n - 1, ... 0 n, n, n, ... n
for any number of rows n.
Does this matrix have a name? I wasn't able to find one.
The eigenvalues of this matrix are 1, 2, 3, ..., n. Is there a simple formula for the eigenvectors? Some experiments seem to show that the first one has elements
(-1)^(k + 1)*k/(k - 1)! for k from 1 to n. But I wasn't able to puzzle out a formula for the others.
It would be easy enough to compute the eigenvectors for any specific value of n, but I am guessing there's a relatively simple general formula, which I've been too lazy to work out for myself. Thanks for any light you can shed on this problem.