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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!

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Looks quite like a generalized Sylvester equation. –  J. M. Jun 9 '12 at 16:01

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