Suppose we have a vector bundle $E$ over a smooth manifold $M$, and let $F$ be a vector bundle endomorphism $E$.
Is there anyway to now generalize the theory of eigenspaces in linear algebra to vector bundles? In particular, I am wondering when it is possible to construct sub bundles which correspond to some family of “eigenvalues” of $F$, and whose Whitney sum is isomorphic to $E$.
In the case where $F$ restricts to a linear isomorphism, which in each fibre has eigenvalues $\pm 1$, it’s not very difficult to construct an isomorphism:
$E\leftrightarrow E^+\oplus E^-$
But trying to generalize this to multiple constant eigenvalues seems difficult, and generalizing this further to non constant eigenvalues seems very difficult.
For context, I am trying to show that every manifold that admits a pseudo-riemannian metric of signature $(t,s)$ splits as a Whitney sum of a positive definite, and negative definite subbundle, and I have an idea that hinges on this generalization. I have unfortunately found no source which either $a)$ proves this statement (though I suspect it to be true) , or $b)$ goes into such a generalization.
If anyone has any reference suggestions, or can provide some help on the path to this generalization it would be of much help.
The issue with the argument you're trying to make is that, without additional restrictions, the eigenspaces of a vector bundle endomorphism can merge, split, and change dimension in complicated ways not easily described using the language of vector bundles. Luckily, there are other ways to achieve the decomposition you describe. I'll address only the positive definite case; the negative definite argument is essentially identical.
Some linear algebra:
Let $\mathbb{R}^{p,q}$ be $\mathbb{R}^{p+q}$ equipped with the standard pseudo-inner product of signature $(p,q)$, and let $N\subseteq\mathbb{R}^{p,q}$ be the negative definite subspace spanned by the last $q$ coordinates.
Let $\operatorname{Gr}(p,\mathbb{R}^{p,q})$, denote the Grasmannian of rank $p$ (that is, the set of $p$-dimensional subspaces equipped with the natural smooth manifold structure). The set of positive definite subspaces is a subset which I'll call $\Delta\subseteq\operatorname{Gr}(p,\mathbb{R}^{p,q})$. Note that each element of $\Delta$ is complimentary to $N$. Thus, each element of $\Delta$ may be written as $\{(\vec{x},A\vec{x}):\vec{x}\in\mathbb{R}^p\}$ where $A$ is a $q\times p$ matrix. Further, each $q\times p$ matrix $A$ corresponds to a unique subspace, which is an element of $\Delta$ iff $\|A\|_{op}<1$. This gives us a diffeomorphism from $\Delta$ to $\{A\in\mathbb{R}^{q\times p}:\|A\|_{op}<1\}$, which is a smoothly contractible open subset of $\mathbb{R}^{q\times p}$.
Vector Bundles
Given a rank $p+q$ smooth vector bundle $\pi:E\to M$ with a bundle pseudo-metric $g$ of signature $p,q$, there is a corresponding bundle $\pi_{\Delta}:E_\Delta\to M$ where the fiber over $x\in M$ is the set of positive definite subspaces of $\pi^{-1}(x)\subseteq E$. The typical fiber of $\pi_\Delta$ is of course just $\Delta$ from above. Since the typical fiber is contractible, $\pi_{\Delta}$ admits a smooth global section (see this mathoverflow post for more discussion). These sections are exactly the subbundles that you're looking for.