157 research outputs found
Factorization of Matrices of Quaternions
We review known factorization results in quaternion matrices. Specifically,
we derive the Jordan canonical form, polar decomposition, singular value
decomposition, the QR factorization. We prove there is a Schur factorization
for commuting matrices, and from this derive the spectral theorem. We do not
consider algorithms, but do point to some of the numerical literature.
Rather than work directly with matrices of quaternions, we work with complex
matrices with a specific symmetry based on the dual operation. We discuss
related results regarding complex matrices that are self-dual or symmetric, but
perhaps not Hermitian.Comment: Corrected proofs of Theorem 2.4(2) and Theorem 3.
Almost commuting unitary matrices related to time reversal
The behavior of fermionic systems depends on the geometry of the system and
the symmetry class of the Hamiltonian and observables. Almost commuting
matrices arise from band-projected position observables in such systems. One
expects the mathematical behavior of almost commuting Hermitian matrices to
depend on two factors. One factor will be the approximate polynomial relations
satisfied by the matrices. The other factor is what algebra the matrices are
in, either the matrices over A for A the real numbers, A the complex numbers or
A the algebra of quaternions.
There are potential obstructions keeping k-tuples of almost commuting
operators from being close to a commuting k-tuple. We consider two-dimensional
geometries and so this obstruction lives in KO_{-2}(A). This obstruction
corresponds to either the Chern number or spin Chern number in physics. We show
that if this obstruction is the trivial element in K-theory then the
approximation by commuting matrices is possible.Comment: 33 pages, 2 figures. In version 2 some formulas have been corrected
and some proofs have been rewritten to improve the expositio
Pencils of complex and real symmetric and skew matrices
AbstractThis expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices. The treatment is in the spirit of the well-known book of Gantmacher on matrix theory, and may be regarded as a supplement to Gantmacher's chapters on pencils of matrices
Lectures on Mechanics
Publisher's description: The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule
Secant varieties of P^2 x P^n embedded by O(1,2)
We describe the defining ideal of the rth secant variety of P^2 x P^n
embedded by O(1,2), for arbitrary n and r at most 5. We also present the Schur
module decomposition of the space of generators of each such ideal. Our main
results are based on a more general construction for producing explicit matrix
equations that vanish on secant varieties of products of projective spaces.
This extends previous work of Strassen and Ottaviani.Comment: 21 page
- …