There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let $\mathcal{E}$ be a vector bundle of rank $r$, with $\mathcal{O}\_X$ the usual sheaf of functions on $X = \mathbb{P}^1$, then we can write our line bundles as the invertible sheaves $\mathcal{O}\_{X}(n)$ with $n \in \mathbb{Z}$. Thus, the decomposition can be stated as $$\mathcal{E} \cong \oplus\_{i=1}^r \mathcal{O}(n\_i) \quad n\_1 \leq \dotsb \leq n\_r.$$


If we use the usual open cover of $\mathbb{P}^1$ with two affine lines $U\_0 = \mathbb{P}^1 - \{\infty\}$ and $U\_1 = \mathbb{P}^1 - \{0\}$, note that $\mathcal{O}\_{U\_0 \cap U\_1} = k[x,x^{-1}]$ (with $\mathcal{O}\_{U\_0} = k[x]$ and $\mathcal{O}\_{U\_1} = k[x^{-1}]$).


A vector bundle (up to isomorphism) $\mathcal{E}$ of rank $r$ is then a linear automorphism on $\mathcal{O}\_{U\_0 \cap U\_1}^r$ modulo automorphisms of each $\mathcal{O}\_{U\_i}^r$ for $i = 0,1$. (I am looking at the definition given in Hartshorne II.5.18 where $A = k[x,x^{-1}]$, the linear automorphisms are $\psi\_1^{-1} \circ \psi\_0$ where $\psi\_i: \mathcal{O}\_{U\_i}^r \rightarrow \left.\mathcal{E}\right\rvert\_{U\_i}$ are isomorphisms, and the definition of isomorphism of vector bundles allows us to change bases of $\mathcal{O}\_{U\_i}^r$.


Thinking of this in linear algebra terms, these linear automorphisms on $\mathcal{O}\_{U\_0 \cap U\_1}^r$ are elements of $GL\_r(k[x,x^{-1}])$, and changing coordinates in $\mathcal{O}\_{U\_i}^r$ are elements of $\operatorname{GL}\_r(k[x])$ for $i = 0$ and $\operatorname{GL}\_r(k[x^{-1}])$ for $i = 1$. Thus up to isomorphism, the vector bundles of rank $r$ on $\mathbb{P}^1$ are elements of the double quotient $$ \operatorname{GL}\_r(k[x^{-1}]) \left\backslash \large{\operatorname{GL}\_r(k[x,x^{-1}])} \right/ \operatorname{GL}\_r(k[x]).$$ The decomposition of vector bundles into line bundles SHOULD mean that these double cosets can be represented by matrices of the form $$\begin{pmatrix} x^{n\_1} & & & 0 \\ & x^{n\_2}& & \\ & & \ddots & \\ 0 & & & x^{n\_r}\end{pmatrix} \quad n\_1 \leq n\_2 \leq \dotsb \leq n\_r.$$ I want to know whether there is a way to prove this fact purely via linear algebra (equivalently, if the geometric proof [cf. Lemma 4.4.1 in Le Potier's "Lectures on Vector Bundles"] has a linear algebraic interpretation).


[Note: For the affine case, taking the double quotient $$\operatorname{GL}\_n(k[x]) \left \backslash M\_{n,m}(k[x]) \right/ \operatorname{GL}\_m(k[x])$$ gives the classification of vector bundles over $\mathbb{A}^1\_k$ (and of course, when replacing $k[x]$ with an arbitrary PID, gives the usual structure theorem of finitely generated modules over PID).]



Using linear algebra to classify vector bundles over ℙ¹
I must admit I have never read this reference, but I remember it from a similar discussion on a German forum, according to which there is a simple proof in


[Michiel Hazewinkel and Clyde Martin, *A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line*, Journal of pure and applied algebra 25 (1982), pp. 207 - 211](http://www.sciencedirect.com/science/article/pii/0022404982900378).


