Given a system of equations, I'm curious whether I can find the closed form solution for $P$,
Here, $G$,$H$ are known $N \times N$ matrix, $\lambda$ is a known scalar;
$s$,$t$ are two $N \times 1$ unknown vectors. $\Lambda$ is an unknown $N \times N$ matrix
$1_{N}$ denotes a $N \times 1$ all one vector (i.e $[1,1,\dots,1]^T$).
$P$ is an unknown $N \times N$ matrix. Here we want to solve P
Suppose we know: $ 2(GPH^T + G^TPH) = (2\lambda I+\Lambda+\Lambda^T)P+1_N s^T+t1_{N}^T$ $ P \cdot 1_N = 1_N $ $ P^T\cdot 1_N = 1_N$ $ PP^T = I$
Here $I$ denotes an $N \times N$ identity matrix
Can anyone gives me some suggestions on this problems? or any books relate to this topic recomended?
If a closed form solution may be impossible, can I get a approximately value for P?
Thanks a lot!