I am trying to understand Example 1.13 of Hatcher's book on vector bundles and K-Theory (page 24).
The cannonical line bundle $H \to \mathbb{C}P^1$ satisfies the relation $(H \otimes H)\oplus 1 \simeq H \oplus H$.
The total space, $H$, is given by $H = ((x,v) \in \mathbb{C} P^1 \times \mathbb{C}^2:v \in X \},$ with the projection map $(x,v) \mapsto x$
To prove this, I need to understand clutching functions. The following is from Atiyah's book 'K-Theory':
Suppose we have a space $X = X_1 \cup X_2$ such that $X_1 \cap X_2 = A$. Assume also we have vector bundles $p_i:E_i \to X_i$ and that $\phi:E_1|A \to E_2|A$ is an isormophism.
Then, we can form a vector bundle $E_1 \cup_\phi E_2 \to X$.
Alternatively, Hatcher defines a map $f:A \to GL_n(\mathbb{C})$ (although he specifically constructs clutching functions from spheres, not arbitrary spaces). The two definitions are equivalent.
In particular we can construct a clutching function over the complex line bundle $\mathbb{C} P^1 \simeq S^2$. This is example 1.10 of Hatcher's book. I am not 100% confident in the argument, but I think I understand it. Regardless, the clutching function derived is $f(z)=(z)$ (i.e. multiplication by $z$)
Now I start to get hazy, in Example 1.13. To quote verbatim:
Let us show that the canonical line bundle $H \to \mathbb{C}P^1$ satisfies the relation $(H \otimes H)\oplus 1 \simeq H \oplus H$ where 1 is the trivial one-dimensional bundle. This can be seen by looking at the clutching functions for these two bundles, which are the maps $S^1 \to GL_2(\mathbb{C})$ given by
$z \mapsto \begin{pmatrix} z^2 & 0 \\ 0 & 1 \end{pmatrix} $
and
$z \mapsto \begin{pmatrix} z & 0 \\ 0 & z \end{pmatrix} $
I get lost in this last argument. I can buy the clutching function for the identity is the second matrix, but where did the first come from? Why do we work over $GL_2(\mathbb{C})$? (In Example 1.10 we worked over $GL_1(\mathbb{C})$.
Moreover, even if I believe these clutching constructions, how can they be used to show that $(H \otimes H)\oplus 1 \simeq H \oplus H$?
Any advice, or references appreciated
Update: Theo's comments below, made me realise I was way off with my thinking. I think the following makes some sense.
Firstly given we have the clutching function for $(z)$ for $H$, then the clutching function for $H \otimes H$ is $(z^2)$. The clutching function for the trivial bundle is the identity.
I am slighly confused as to why we can then take the direct sum of the bundles as the matrix with components along the diagonal. My best thought is that there is a group homomorphism $GL(m,\mathbb{C}) \times GL(n,\mathbb{C}) \to GL(m+n,\mathbb{C})$ given by $\Theta(A,B) = \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix} $
From this, it is clear how to construct the clutching functions above. Hatcher gives the general homotopy via a path $\alpha_t \in GL_{2n}(\mathbb{C})$ from the identity matrix, to the matrix of the transformation which interchanges the two factors of $\mathbb{C}^n \times \mathbb{C}^n$