I am trying to find the condition on $K$ under which the matrix $$ M=\begin{bmatrix} K-P &KWL^c\\ L^cWK & L^cWKWL^c \end{bmatrix} $$ is positive semidefinite. Where $L^c$ is Laplacian Matrix, and $$ K=\begin{bmatrix} k_1&0&0\\0&k_2&0\\0&0&k_3 \end{bmatrix} $$ $$ P=\begin{bmatrix} p_1&0&0\\0&p_2&0\\0&0&p_3 \end{bmatrix} $$ are positive-definite.

PS: I have tried the following $$ \underbrace{\begin{bmatrix} K-P &KWL^c\\ L^cWK & L^cWKWL^c \end{bmatrix}}_{M}= \underbrace{\begin{bmatrix} I &O\\ O & L^cW \end{bmatrix}}_{N} \underbrace{\begin{bmatrix} K-P &K\\ K & K \end{bmatrix}}_{S} \underbrace{\begin{bmatrix} I &O\\ O &WL^c \end{bmatrix}}_{N^{\top}} $$ If $S$ is positive definite then $M$ is positive-semidefinite. But $S$ can never be positive definite (One can verify this using schurs complement analysis). Moreover, this only a sufficient condition as $N$ is not a full rank Matrix.


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