2
votes
1answer
63 views

under what conditions a product of matrices is the identity matrix (more complicated than that)?

I have a set of matrices $A_1,\ldots,A_n$ and another set $B_i = A_i^{-1}$ for $i = 1,\ldots, n$ (I assume $A_i$ are invertible). Let $\mathcal{A} = \{A_i\} \cup \{B_i\}$. What are some simple ...
1
vote
1answer
165 views

Semigroups of matrices with zeroes and a single 1

I stumbled upon this while reviewing a Harvard lecture on abstract algebra. What I want to know is if these semigroups are known and, if so, what they are called. I've checked the assertions below for ...
2
votes
2answers
512 views

The compactness of the unit sphere in finite dimensional normed vector space

We define $ (\mathbb{R}^m, \|.\|)$ to be a finite dimensional normed vector space with $ \|.\|$ is defined to be any norm in $ \mathbb{R^m}$. Let $S = \lbrace x \in \mathbb{R}^m: \| x\| = 1 \rbrace.$ ...