Simplification of certain linear matrix equations

It is shown that in solving the equation
\Sigma \alpha_{ij}(A^{T})^{i}PA^{j} = Q
for the symmetric matrix
P
the number of linear equations and unknowns call be reduced from
\frac{1}{2} n(n + 1)
to
\frac{1}{2} n(n-1)
by introducing a skew symmetric matrix. This generalizes earlier results for Lyapunov matrix equations.