On an infinite family of unipotent Sylvester–Kac-like matrices

Classical Sylvester–Kac matrices are tridiagonal integral matrices with positive off-diagonal entries and fully integral spectra. Here, by relaxing the positivity requirement and using a lower Pascal triangle framework, we define, for each positive integer n, a unipotent Sylvester–Kac-like matrix in which n is the only eigenvalue. This construction highlights the connection to the original Sylvester–Kac matrices while introducing a new family of unipotent matrices with distinctive properties.