3
votes
1answer
147 views

Proof that $U(n)$ is connected

I'm trying to prove that $U(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I\}$ is connected, but most of the proof comes down to proving that $SU(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I $ and $ ...
1
vote
2answers
116 views

$GL_n(\mathbb{R})$ is the union of two path connected subsets (Michael Artin's Algebra).

I have that $SL_n(\mathbb{R})$ is path connected and generated by elementary matrices of the form $I + a e_{i,j} (i \neq j)$, where $e_{i,j}$ is the matrix with $1$ at position $(i,j)$ and zero ...
5
votes
2answers
173 views

Is the set $ {\rm SL }(n, \mathbb {R}) $ connected in ${\rm M }(n, \mathbb {R}) $

Is the set $ {\rm SL }(n, \mathbb {R}) $ connected in ${\rm M }(n, \mathbb {R}) $? Can you give me hints? I have only concept on following tags.
21
votes
4answers
693 views

Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times ...
3
votes
1answer
168 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 ...
12
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 ...
10
votes
3answers
305 views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...