  In a recent article "Projective geometries, $Q$-polynomial structures, and
quantum groups" Terwilliger (arXiv:2407.14964) defined a certain weighted
adjacency matrix, depending on a free (positive real) parameter, associated
with the projective geometry, and showed (among many other results) that it is
diagonalizable, with the eigenvalues and their multiplicities explicitly
written down, and that it satisfies the $Q$-polynomial property (with respect
to the zero subspace).
  In this note we
  (i) Write down an explicit eigenbasis for this matrix.
  (ii) Evaluate the adjacency matrix-eigenvector products, yielding a new proof
for the eigenvalues and their multiplicities.
  (iii) Evaluate the dual adjacency matrix-eigenvector products and directly
show that the action of the dual adjacency matrix on the eigenspaces of the
adjacency matrix is block-tridiagonal, yielding a new proof of the
$Q$-polynomial property.
