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For $H \subset G$, write $e_H$ for the basis element of $B(G)$ corresponding to the $G$-set $G/H$. Recall that for $K \subset G$, $$ e_H\cdot e_K = \sum\limits_{H g K \in H \backslash G / K} e_{H \cap g K g^{-1}}. $$ By partially ordering the $e_H$ with respect to the subconjugacy partial ordering on the set of conjugacy classes of subgroups of $G$, we have that $M_{e_H}$ is lower-triangular for each $H \subset G$. The eigenvalues are then the diagonal elements. The eigenvalue corresponding to $e_G$ is $|G/H|$. The eigenvalue corresponding to $e_K$ for $K \subset G$ is bounded above by the number of $(H, K)$-double cosets in $G$, which is bounded above by $[G:H] = |G/H|$.