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I need to know whether the $1)$ The set of all symmetric positive definite matrices are connected or not?

Well I guess, This set is convex set, Let $M$ be a symmetric positive definite so $X^TMX>0, X\in \mathbb{R}^n$, now for any two such matrix $A,B$, we have $X^T[tA+(1-t)B]X=tX^TAX+(1-t)X^TBX>0, t\in[0,1]$ Hence This set is path connected.

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Correct ${}{}{}$ – Will Jagy Oct 25 '12 at 5:37

It's correct (you just have to specify you take $X\neq 0$).

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