A Canonical Form for Real Matrices under Orthogonal Transformations

Szegő and para-orthogonal polynomials on the real line: Zeros and canonical spectral transformations

We study polynomials which satisfy the same recurrence relation as the Szegő polynomials, however, with the restriction that the (reflection) coefficients in the recurrence are larger than one in modulus. Para-orthogonal polynomials that follow from these Szegő polynomials are also considered. With positive values for the reflection coefficients, zeros of the Szegő polynomials, para-orthogonal ...

Crossing Matrices and Thurston's Canonical Form for Braids

Parameterization of real orthogonal antisymmetric matrices

In response to a question posed by João P. Silva, I demonstrate that an arbitrary 2n×2n real orthogonal antisymmetric matrix can be parameterized by n(n−1) continuous angular parameters. An algorithm is provided for constructing the general form for such a matrix. This algorithm is based on observation that Sp(n,R) ∩ O(2n) ∼= U(n) and employs the parameterization of the coset space SO(2n)/U(n)....

Ela Change of the *congruence Canonical Form of 2-by-2 Matrices under Perturbations

It is constructed the Hasse diagram for the closure ordering on the sets of *congruence classes of 2× 2 matrices. In other words, it is constructed the directed graph whose vertices are 2× 2 canonical complex matrices for *congruence and there is a directed path from A to B if and only if A can be transformed by an arbitrarily small perturbation to a matrix that is *congruent to B.

Extended covariance under nonlinear canonical transformations in Weyl quantization

A theory of non-unitary-invertible as well as unitary canonical transformations is formulated in the context of Weyl's phase space representations. Exact solutions of the transformation kernels and the phase space propagators are given for the three fundamental canonical maps as linear, gauge and contact (point) transformations. Under the nonlinear maps a phase space representation is mapped to...