Consider the set of all $n\times n$ matrices with real entries, considered as the space $\mathbb{R^{n^2}}$ What can we say about connectedness and compactness of the following sets?
- The set of all orthogonal matrices.
- The set of all matrices with trace equal to unity.
- The set of all symmetric and positive definite matrices.
I need help to understand basic concepts of solving these kind of problems. I mean how to show connectedness or compactness when we have set of matrices with certain specific properties. As mentioned above. Thanks