Hoffman Linear Algebra 1971

These notes were written as a part of a graduate level course on transform theory offered at King's College London during 2002 and 2003. The material is heavily indebt to the excellent textbook by Gilbert Strang [1], which the reader is referred to for a more complete description of the material; for a more in-depth coverage, the reader is referred to [2–6].

The purpose of these lectures to report on the recent solution of a 50 years old problem of describing the set of the eigenvalues of a sum of two hermitian matrices with prescribed eigenvalues 1 Statement of the problem For a field F denote by F n the vector space of column vectors f = (f 1 , ..., f n) T with entries in F. We will mostly assume that F is either the field of reals R or complexes C. We view R n and C n as inner product spaces with the inner product (x, y) equal to either y T x or y * x respectively. Set R n ≥ := {x = (x 1 , ..., x n) T ∈ R n : x 1 ≥ x 2 ≥ • • • ≥ x n }. Let S n ⊂ H n be the real vector spaces of n × n real symmetric and hermitain matrices respectively. Note that S n and H n describe the space of selfadjoint operators in R n and C n respectively, with respect to the standard inner product (•, •). Let A ∈ H n. It is well known that C n has an orthonormal basis consisting entirely of the eigenvectors of A:

Linear Algebra and its Applications, 1988

This book is the rewritten second edition of the very successful textbook by Peter Lancaster of the same title.

Integral Equations and Operator Theory, 1989

2012

The goal of this chapter is to show that there are nice normal forms for symmetric matrices, skew-symmetric matrices, orthogonal matrices, and normal matrices. The spectral theorem for symmetric matrices states that symmetric matrices have real eigenvalues and that they can be diagonalized over an orthonormal basis. The spectral theorem for Hermitian matrices states that Hermitian matrices also have real eigenvalues and that they can be diagonalized over a complex orthonormal basis.

Global Pseudo-Differential Calculus on Euclidean Spaces, 2010

Linear algebra has two aspects. Abstractly, it is the study of vector spaces over fields, and their linear maps and bilinear forms. Concretely, it is matrix theory: matrices occur in all parts of mathematics and its applications, and everyone working in the mathematical sciences and related areas needs to be able to diagonalise a real symmetric matrix. So in a course of this kind, it is necessary to touch on both the abstract and the concrete aspects, though applications are not treated in detail.

Contemporary Mathematics, 2006

We formulate the concept of unique normal form in terms of a spectral sequence. As an illustration of this technique we reproduce some results of Baider and Churchill concerning the normal form of the anharmonic oscillator. The aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory

Transactions of the American Mathematical Society, 1987

If G is a compact Lie group and M a Riemannian G-manifold with principal orbits of codimension k then a section or canonical form for M is a closed, smooth k-dimensional submanifold of M which meets all orbits of M orthogonally. We discuss some of the remarkable properties of G-manifolds that admit sections, develop methods for constructing sections, and consider several applications.