Solving lyapunov equation, Matlab has different solution, why? I need to solve the lyapunov equation i.e. $A^TP + PA = -Q$. With $A = \begin{bmatrix} -2 & 1 \\ -1 & 0 \end{bmatrix}$ and $Q = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$. 
Hence...
$$
\begin{bmatrix} 
  -2 & -1 \\
   1 &  0
\end{bmatrix}
\begin{bmatrix}
P_{11} & P_{12} \\
P_{12} & P_{22}
\end{bmatrix}
+
\begin{bmatrix}
P_{11} & P_{12} \\
P_{12} & P_{22}
\end{bmatrix}
\begin{bmatrix}
-2 & 1 \\
-1 & 0
\end{bmatrix} + 
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix} = 0
$$
$$
\begin{bmatrix}
-4P_{11} - 2P_{12} + 1 & P_{11} -2P_{12} - P_{22} \\
P_{11} - 2P_{12} - P_{22} & 2P_{12} + 1
\end{bmatrix} = 0
$$
$$
\begin{bmatrix}
-4 & -2 & 0 \\
1 & -2 & -1 \\
1 & -2 & -1 \\
0 & 2 & 0 
\end{bmatrix}
\begin{bmatrix}
P_{11} \\
P_{12} \\
P_{22}
\end{bmatrix} =
\begin{bmatrix}
-1 \\
0 \\
0 \\
-1 
\end{bmatrix}
\Rightarrow 
\begin{bmatrix}
-4 & -2 & 0 \\
1 & -2 & -1 \\
0 & 2 & 0 
\end{bmatrix}
\begin{bmatrix}
P_{11} \\
P_{12} \\
P_{22}
\end{bmatrix} =
\begin{bmatrix}
-1 \\
0 \\
-1 
\end{bmatrix}$$
And such we get that $P = \begin{bmatrix} 1/2 & -1/2 \\ -1/2 & 3/2\end{bmatrix}$.
This is the same solution as given by my professor. I wanted to check however if I can also find the solution using Matlab. I entered the following:
A = [-2 1; -1 0];
Q = [1 0; 0 1];
P = lyap(A,Q)

This however tells me that $P = \begin{bmatrix} 1/2 & 1/2 \\ 1/2 & 3/2\end{bmatrix}$.
What is going on here? Is Matlab correct or wrong? Or is my solution wrong? Or are we both correct?
 A: The matlab definition must be different from yours. With Matlab's $P$, one has $AP+PA^T=-Q$. Doing things your way with matlab's $P$ gives 
$$
A^TP+PA=
\begin{bmatrix}
-3 & -2 \\
-2 & 1
\end{bmatrix}
$$
Maybe look in matlab's documentation to see what equation their lyap(A,Q) is solving; I'd guess it had the transpose switched.
EDIT: It seems Mathematica9 has the other $AP+PA^T=-Q$ definition, or close to it.
Their command LyapunovSolve[a,c] gives solution to $ax+xa^T=c$, which has the transposed matrix mentioned last, opposite to your version. Reference page:
http://reference.wolfram.com/mathematica/ref/LyapunovSolve.html
A: X = lyap(A,B,-C) solves the continuous-time Sylvester equation
AX + XB = C
and 
X = lyap(A’,Q) solves the continuous-time Lyapunov equation
ATP + PA + Q = 0
so, you can solve the lyapunov function.
A = [-2 1; -1 0];
Q = [1 0; 0 1];
P = lyap(A',Q)
P = [0.5000    -0.5000]
    [-0.5000    1.5000]
A: from MATLAB documents it actually solves $AP+PA'+Q=0$ so that the result is different than your equation $A'P+PA+Q=0$
to get the same answer in MATLAB use $A'$ instead of $A$ since $A'P+P(A')'+Q=0$ equals
$A'P+PA+Q=0$ which is equivalent to your equation. eventually the following 
lyap(A',Q) matches your results
