Recently I have faced two questions, which are solved in this website. One is to show that the space of orthogonal matrices is compact, another one is to show that the space of nilpotent matrices is connected.
I want to study and solve problems like these on topological aspects on the set of matrices, but unable to find any book or notes specially on this topic.
It will be beneficial if someone can suggest some materials.
Thanks in advance!!
What you might want to do is to familiarise yourself with the properties of $S_1\cap S_2\cap\cdots\cap S_k$,
where each $S_i$ is of the form $p(x_1,x_2,\ldots,x_{n^2})=0$ or of the form $p(x_1,x_2,\ldots,x_{n^2})\leq0$.
Here, the variables $x_j$ are supposed to represent the entries of a matrix.
Sets of the above two forms are the object of study of algebraic geometry.
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Alternatively, $GL_n(\mathbb{R})$ i.e. the set of invertible real matrices is an example of a topological group. Topological groups are themselves an enormous subject.
