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Given a set $M$ of all $n\times n$ real matrices with the usual norm topology. Then, is the set of all symmetric positive definite matrices in $M$ connected?

Also, is the set of all invertible matrices in $M$ compact?

All I can infer so far is that the set $M$ is probably not compact as, if $M$ is positive definite then so is $cM$ for positive scalars $c$ and as $c\rightarrow \infty$, there is no limit.

Thank You.

The Doctor
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Yes, the set of positive definite, symmetric matrices is connected. Both the positive definite matrices and the symmetric matrices form convex sets, the intersection of convex sets is convex, and convex sets are connected. You can verify that these properties are convex directly from the definitions without too much work.

The invertible matrices are not compact. An easy proof is that the continuous image of a compact set is compact. Therefore, if the invertible matrices were compact, there would have to be a bound on their determinants. (You didn't ask, but continuity of the determinant also shows that the invertible matrices are not connected: if they were, the set of determinants of invertible matrices would be connected.)

  • OP is asking about the set of invertible matrices in $M$, not all invertible matrices. Nevertheless, this subset is equal to $M$ since every positive definite matrix has nonzero determinant, and as OP noted above, $M$ is not compact. – silvascientist Oct 08 '17 at 04:54
  • @MonstrousMoonshiner I thought $M$ was the set of all matrices, and the set of positive definite matrices was unnamed? We'll let OP clarify. –  Oct 08 '17 at 04:55
  • After rereading the question, it seems OP is making conflicting usages of $M$. He defines $M$ at the start as being the set of all matrices, then uses $M$ later to refer to the symmetric positive definite matrices. I think that OP wants the invertible matrices in the second $M$, but we definitely need clarification. – silvascientist Oct 08 '17 at 04:58