2
votes
2answers
66 views

Naive question about the group $SU(n)$?

As usual, let $SU(n)$ represent the set of all the $n\times n$ unitary matrices with determinant $1$. It's easy to show that any matrix $U$ takes the form $U=e^{iA}$ ($A$ is a $n\times n$ traceless ...
3
votes
3answers
162 views

Which of the following subsets of $M_n(\mathbb{R})$ are compact (NBHM)

Following is a list of problems from an exam for admission into Ph. D program. I have just compiled all previous questions on compactness of certain subsets of matrices and i tried to work out . I ...
1
vote
1answer
114 views

Prove the following set is compact

$\def\R{\mathbb R}$Fix vectors $b\in\R^k_+$ and $D\in\R^k_{++}$, and a matrix $A\in\R^{N\times k}$. Here, $\R^k_+$ denotes the set of vectors in $\R^k$ whose entries are nonnegative, and $\R^k_{++}$ ...
3
votes
1answer
185 views

Topological properties of symmetric positive definite matrices

Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true? (a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$. (b) $S$ is ...
2
votes
1answer
63 views

$GL(2,\mathbb{R})$ as a subset of $\mathbb{R}^4$

If we consider $GL(2,\mathbb{R})$ as a topological subspace of $\mathbb{R}^4$ with the usual topology and want to know if it compact or not then if we could show that it was not closed then we would ...
13
votes
2answers
2k views

Topology of matrices

1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all ...
3
votes
2answers
204 views

Which of the following are compact sets?

Which of the following are compact sets? $\{\operatorname{trace}(A): A \text{ is real orthogonal}\}$ $\{A\in M_n(\mathbb{R}):\text{ eigenvalues $|\lambda|\le 2$}\}$ Well, orthogonal matrices are ...