Various variants of the matrix $$A_{J,K} = |J\cap K|$$ were studied, and the spectrum was computed. A one-parameter variant is given by $$A^{(i)}_{J,K} = \binom{|J\cap K|}{i}$$ for some fixed $i$, and your problem corresponds to $i=1$. In total, six variants are given in a this friendly paper by Ghareghani, Ghorbani and Mohammad‐Noori. In particular, the spectrum of $A^{(i)}$ is described in Lemma 9. The lemma is attributed to R. M. Wilson (1982), who uses somewhat different notation and terminology.

Other useful references, including to additional works of R. M. Wilson, appear in the bibliography of the G-G-MN paper.

Various variants of the matrix $$A_{J,K} = |J\cap K|$$ were studied, and the spectrum was computed. A one-parameter variant is given by $$A^{(i)}_{J,K} = \binom{|J\cap K|}{i}$$ for some fixed $i$, and your problem corresponds to $i=1$. In total, six variants are given in Section 3 of this friendly paper by Ghareghani, Ghorbani and Mohammad‐Noori. In particular, the spectrum of $A^{(i)}$ is described in Lemma 9. The lemma is attributed to R. M. Wilson (1982), who uses somewhat different notation and terminology. The case $i=1$ recovers your theorem, and in general the non-zero eigenvalues are given by $$\lambda_j=\binom{s-j}{i-j}\binom{n-i-j}{s-i}, \qquad j=0,1,2,\ldots,i,$$ with $\lambda_j$ having multiplicity $\binom{n}{j}-\binom{n}{j-1}$ (note that $\binom{n}{-1}:=0$). The eigenvalue 0 has multiplicity equal to the remaining dimension, i.e. $N-\binom{n}{i}$.

Other useful references, including to additional works of R. M. Wilson, appear in the bibliography of the G-G-MN paper.