Let $g$ be an $n \times n$ matrix of functions $g_{ij}(z)$ in $\mathbb{C}(z)$. Suppose that the $g_{ij}(z)$ have no poles on the annulus $1-\epsilon < |z| < 1+\epsilon$ and that $\det g(z)$ is nonzero on this annulus. Then I can define a holomorphic line bundle on $\mathbb{P}^1$ by taking trivial rank $n$ bundles on $\{ |z|<1+\epsilon \}$ and $\{ |z| > 1-\epsilon \}$ and gluing them by $g$. This bundle must be isomorphic to $\bigoplus \mathcal{O}(d_i)$ for some unique $d_1 \geq d_2 \geq \cdots \geq d_n$.

How do I read off the $d_i$ from the $g_{ij}(t)$?

For comparison, I know a simple criterion when the annulus is replaced with all of $\mathbb{C} \setminus \{ 0 \}$: $d_1+d_2+\cdots + d_k$ is the minimum, over all $k \times k$ minors of $g$, of the order of vanishing of that minor at $0$.

Let $g$ be an $n \times n$ matrix of functions $g_{ij}(z)$ in $\mathbb{C}(z)$. Suppose that the $g_{ij}(z)$ have no poles on the annulus $1-\epsilon < |z| < 1+\epsilon$ and that $\det g(z)$ is nonzero on this annulus. Then I can define a holomorphic vector bundle on $\mathbb{P}^1$ by taking trivial rank $n$ bundles on $\{ |z|<1+\epsilon \}$ and $\{ |z| > 1-\epsilon \}$ and gluing them by $g$. This bundle must be isomorphic to $\bigoplus \mathcal{O}(d_i)$ for some unique $d_1 \geq d_2 \geq \cdots \geq d_n$.

How do I read off the $d_i$ from the $g_{ij}(t)$?

For comparison, I know a simple criterion when the annulus is replaced with all of $\mathbb{C} \setminus \{ 0 \}$: $d_1+d_2+\cdots + d_k$ is the minimum, over all $k \times k$ minors of $g$, of the order of vanishing of that minor at $0$.