  Hessenberg decomposition is the basic tool used in computational linear
algebra to approximate the eigenvalues of a matrix. In this article, we
generalize Hessenberg decomposition to continuous matrix fields over
topological spaces. This works in great generality: the space is only required
to be normal and to have finite covering dimension. As applications, we derive
some new structure results on self-adjoint matrix fields, we establish some
eigenvalue separation results, and we generalize to all finite-dimensional
normal spaces a classical result on trivial summands of vector bundles.
Finally, we develop a variant of Hessenberg decomposition for fields of bounded
operators on a separable, infinite-dimensional Hilbert space.
