All Questions

Let $M_n(\mathbb{C})$ represent the space of $n \times n$ matrices over $\mathbb{C}$. We will think of it as a $\mathbb{C}$-vector space. Notice that if $A \in M_n(\mathbb{C})$ is invertible, then the ...

Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...

Let $p$ be an odd prime, and consider the group $$\{U\in \operatorname{SL}_n(\mathbb{Z}/p\mathbb{Z}) : U^{t}U=I \bmod p \}\subseteq \operatorname{SL}_n(\mathbb{Z}/p\mathbb{Z}).$$ I wonder what is the ...

Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?