74 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.
Principal angles and approximation for quaternionic projections
We extend Jordan's notion of principal angles to work for two subspaces of
quaternionic space, and so have a method to analyze two orthogonal projections
in M_n(A) for A the real, complex or quaternionic field (or skew field). From
this we derive an algorithm to turn almost commuting projections into commuting
projections that minimizes the sum of the displacements of the two projections.
We quickly prove what we need using the universal real C*-algebra generated by
two projections.Comment: 11 pages, 4 figures, 4 auxiliary Matlab file
Quantitative K-Theory Related to Spin Chern Numbers
We examine the various indices defined on pairs of almost commuting unitary
matrices that can detect pairs that are far from commuting pairs. We do this in
two symmetry classes, that of general unitary matrices and that of self-dual
matrices, with an emphasis on quantitative results. We determine which values
of the norm of the commutator guarantee that the indices are defined, where
they are equal, and what quantitative results on the distance to a pair with a
different index are possible. We validate a method of computing spin Chern
numbers that was developed with Hastings and only conjectured to be correct.
Specifically, the Pfaffian-Bott index can be computed by the "log method" for
commutator norms up to a specific constant
The Point in Weak Semiprojectivity and AANR Compacta
We initiate the study of pointed approximative absolute neighborhood
retracts. Our motivation is to generate examples of C*-algebras that behave in
unexpected ways with respect to weak semiprojectivity. We consider both weak
semiprojectivity (WSP) and weak semiprojectivity with respect to the class of
unital C*-algebras (WSP1). For a non-unital C*-algebra, these are different
properties.
One example shows a C*-algebra can fail to be WSP while its unitization is
WSP. Another example shows WSP1 is not closed under direct sums.Comment: Corrected the statement of Theorem 4.16(b
Estimating Norms of Commutators
We find estimates on the norms commutators of the form [f(x), y] in terms of
the norm of [x, y] assuming that x and y are contractions in a C*-algebra A,
with x normal and with spectrum within the domain of f. In particular we
discuss [x^2, y] and [x^(1/2), y] for 0 <=, x <=, 1. For larger values of
\delta = \|[x; y]\| we can rigorous calculate the best possible upper bound
\|[f(x), y]\| for many f. In other cases we have conducted numerical
experiments that strongly suggest that we have in many cases found the correct
formula for the best upper bound.Comment: We are posting the next version of this paper at :
http://repository.unm.edu/handle/1928/23462. Also posted at
http://repository.unm.edu is theMatlab code used to generate example
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