ChapterZero

Two matrix problems

Filed under: Mathematics — Alex @ 12:43 pm 8/5/2006

I came across these two problems this week; they’re both pretty neat. The first is to show that any positive matrix can be represented as a polynomial in terms of P^2 . The second is to show that if $\left\{U^{(n)} = \left(a_{ij}^{(n)}\right) \right\}$ is a sequence of unitary matrices, then there is a subsequence $\left\{U^{(n_k)}\right\}$ satisfying $a_{ij}^{(n_k)} \rightarrow a_{ij}$ for each i,j , and $U = (a_{ij})$ is unitary. Contrary to first appearances, I think the first one is conceptually trickier than the second; but they’re both essentially one-liners.