# Positive semi-definite Matrix 3

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.