Hamiltonian and Symplectic Tensors in the T-product Algebra
Provides a theoretical framework for Hamiltonian and symplectic structures in third-order tensors, relevant for applications like quantum dynamics, but the results are incremental extensions of known matrix decompositions.
The paper defines Hamiltonian and symplectic tensors in the T-product algebra, derives a constructive T-Williamson normal form for tensors with real symmetric positive-definite Fourier slices, and shows limitations for Hermitian slices. Numerical experiments confirm the construction with O(pn^3) runtime.
We study Hamiltonian and symplectic tensor structures in the T-product algebra. We define T-Hamiltonian and T-symplectic tensors and characterize them through their Fourier-domain slices. For T-Hamiltonian tensors we establish the standard block form and the spectral symmetry of T-eigenvalues, while for T-symplectic tensors we derive the inverse and exponential-map properties. Our main result is a constructive T-Williamson normal form for tensors whose Fourier-domain slices are real symmetric positive-definite matrices. We also show that, under the Hermitian symplectic convention adopted here, this decomposition does not extend directly to arbitrary Hermitian positive-definite Fourier-domain slices, and we derive a real-valued recovery criterion under Fourier conjugate symmetry. Numerical experiments verify the construction, exhibit runtime trends consistent with the slice-wise complexity $O(pn^3)$, and illustrate the framework on a Fourier-domain encoding of covariance-matrix families arising in continuous-variable quantum dynamics.