A continuous-time Lyapunov equation is a matrix equation of the form
 |
(1)
|
where 
and 
are given 
real matrices and 
is unknown. It is a special case of the Sylvester
equation. A unique solution exists for every 
iff no two eigenvalues
of 
,
including a repeated choice of the same eigenvalue,
sum to zero.
An equivalent convention used in linear stability is
 |
(2)
|
If all eigenvalues of 
have negative real parts and
is a symmetric positive
definite matrix, then the unique solution is the symmetric positive
definite matrix
 |
(3)
|
where the exponentials are matrix exponentials. The quadratic form 
is a Lyapunov function
for the system 
,
since
 |
(4)
|
for 
.
The case 
gives Lyapunov's first theorem (Boyd 2009).
The solution of 
for real matrices is implemented in the Wolfram
Language as LyapunovSolve[a,
c]. For complex matrices, this function
uses the conjugate transpose in place of the
transpose.
