Simplification of the Lyapunov matrix equationA_{T}PA - P = -Q

It is shown that in solving for the symmetric matrix
P
the number of linear equations and unknowns can be reduced from
\frac{1}{2}n(n + 1)
to
\frac{1}{2}n(n - 1)
by introducing a skew-symmetric matrix. This corresponds to an earlier result for the equation
A_{1}^{T}P_{1} + P_{1}A = -Q
.