(±1)-Invariant sequences and truncated Fibonacci sequences

Let P=ij,(i,j=0,1,2,…) and D=diag((−1)0, (−1)1, (−1)2, …). As a linear transformation of the infinite dimensional real vector space R∞ = {(x0, x1, x2, …)T ∣ xi ∈ R for all i}, PD has only two eigenvalues 1, −1. In this paper, we find some matrices associated with P whose columns form bases for the eigenspaces for PD. We also introduce truncated Fibonacci sequences and truncated Lucas sequences and show that these sequences span the eigenspaces of PD.