To distinguish among the various subsets of $M_n(\Bbb R)$

Question

I am having problem in doing a certain type of problems relating to matrices:

To distinguish among the various subsets of $M_n(\Bbb R)$ such as symmetric, diagonal, diagonalizable, upper triangular, trace zero, definite types, and many more......which are closed , connected, no where dense, compact in $M_n(\Bbb R)$.

I know that a $n \times n$ matrix can be consider homeomorphic to an element of $R^{n^2}$ and then all the topological properties are related accordingly...But all time it is not possible...

Can anyone please suggest reference books or notes....where these type of problems are discussed..

Thank You..

Did not get an answer here...help please..

Answer 1

Analysis in Euclidean Space by K. Hoffman has plenty of examples and exercises on "matrix analysis," including things like compactness and connectedness of some of the spaces you mentioned. He won't spoon feed you the results, but if you read carefully you should pick up a few useful techniques.