asking for referrence in topological properties of matrices can anyone tell me about some books or references where I can find different topological properties(such as compactness , connectedness , completeness , open , bounded , dense , nowhere dense etc....) of different set of matrices( set of all nxn orthogonal , unitary , determinant 1 , symmetric , hermitian, +ve definite , -ve definite etc.)
 A: Your first three sets of matrices are Lie groups, you fourth set is a simply vector subspace of the finite-dimensional real vector space $M_n(\mathbb{R}) \cong \mathbb{R}^{n^2}$ of real $n \times n$ matrices, and your fifth set is a Lie algebra, and in particular a vector subspace of the finite-dimensional real vector space $M_n(\mathbb{C}) \cong \mathbb{R}^{2n^2}$ of complex $n \times n$ matrices. Finally, the set of all postive semidefinite matrices---studying negative semidefinite matrices is one and the same problem---is a cone in $M_n(\mathbb{R})$ or $M_n(\mathbb{C})$ (whichever you're working with), and in particular is convex.
So, from a geometric and topological standpoint, your first three examples are smooth manifolds whose properties are discussed in the expository literature on Lie groups, your next two examples are just finite-dimensional real vector spaces (indeed, inner product spaces), and your final example (grouping the last two together) is extensively studied in the literature on quantum information.
