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Linear Matrix Inequalities (LMIs)

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A Matrix Inequality for positive definite matrices

The answer is No. Here is a counter-example: $$X=\begin{pmatrix} 9 & 3 \\ 3 & 1 \end{pmatrix},\qquad Y=\begin{pmatrix} 1 & 3 \\ 3 & 9 \end{pmatrix}.$$
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