A simple question about one dimesional projections

Question

Does any one-dimensional projection in matrix $\mathbb{M}_n$ has a fixed form？ and Whether there is a minimal partial isometry $u$ in $\mathbb{M}_n$ such that $u^*u=p,uu^*=q$ for any one-dimensional projections $p,q$ in matrix $\mathbb{M}_n$? please help me!!!

Answer 1

Let $p,q$ be rank one projections, meaning that there are $a,b$ of norm $1$ so that, $p=a\otimes a^*$ $q=b\otimes b^*$ (here $a^*$ is the dual of $a$ via the scalar product). If you write it out you find: $$px=\langle a,x\rangle a\quad qx=\langle b,x\rangle b$$

Then with $u=a\otimes b^*$ you have $u^*=b\otimes a^*$. It follows $$uu^*=a\otimes b^*\cdot b\otimes a^*=b^*(b)\cdot a\otimes a^*=\|b\|^2 p=p$$ similarly $$u^*u=b\otimes a^*\cdot a\otimes b^* =\|a\|^2b\otimes b^*=q$$