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I need to prove this: If A is an n$\times$n positive semidefinite matrix, and S any $n \times m$ matrix, then $S^TAS$ is positive semidefinite.

what_456
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  • What have you tried? Where are you stuck? Have you written down the definition of what it means for $A$ to be positive semidefinite? for $S^\top A S$ to be positive semidefinite? – angryavian May 15 '19 at 17:55

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$x^TS^TASx=(Sx)^TA(Sx)$. Therefore, this quantity is $\ge 0$ by hypothesis on $A$.