  This paper lays the groundwork for the theory of categorical diagonalization.
Given a diagonalizable operator, tools in linear algebra (such as Lagrange
interpolation) allow one to construct a collection of idempotents which project
to each eigenspace. These idempotents are mutually orthogonal and sum to the
identity. We categorify these tools. At the categorical level, one has not only
eigenobjects and eigenvalues but also eigenmaps, which relate an endofunctor to
its eigenvalues. Given an invertible endofunctor of a triangulated category
with a sufficiently nice collection of eigenmaps, we construct idempotent
functors which project to eigencategories. These idempotent functors are
mutually orthogonal, and a convolution thereof is isomorphic to the identity
functor.
  In several sequels to this paper, we will use this technology to study the
categorical representation theory of Hecke algebras. In particular, for Hecke
algebras of type A, we will construct categorified Young symmetrizers by
simultaneously diagonalizing certain functors associated to the full twist
braids.
