Related papers: Block Diagonalization of Quaternion Circulant Matr…
We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we…
In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their Hamilton matrices. After that we investigate commutative…
A selfadjoined block tridiagonal matrix with positive definite blocks on the off-diagonals is by definition a Jacobi matrix with matrix entries. Transfer matrix techniques are extended in order to develop a rotation number calculation for…
We propose a novel factorization of a non-singular matrix $P$, viewed as a $2\times 2$-blocked matrix. The factorization decomposes $P$ into a product of three matrices that are lower block-unitriangular, upper block-triangular, and lower…
We show how to visualize the process of diagonalizing the Hamiltonian matrix to find the energy eigenvalues and eigenvectors of a generic one-dimensional quantum system. Starting in the familiar sine-wave basis of an embedding infinite…
This work proposes double-bracket iterations as a framework for obtaining diagonalizing quantum circuits. Their implementation on a quantum computer consists of interlacing evolutions generated by the input Hamiltonian with diagonal…
We consider the Hamiltonian coupled-mode system that occur in nonlinear optics, photonics, and atomic physics. Spectral stability of gap solitons is determined by eigenvalues of the linearized coupled-mode system, which is equivalent to a…
We present the first deterministic, finite-step algorithm for exact tensor ring (TR) decomposition, addressing an open question about the existence of such procedures. Our method leverages blockwise simultaneous diagonalization to recover…
Iterative Fast Fourier Transform methods are useful for calculating the fields in composite materials and their macroscopic response. By iterating back and forth until convergence, the differential constraints are satisfied in Fourier…
Tensor operations play an essential role in various fields of science and engineering, including multiway data analysis. In this study, we establish a few basic properties of the range and null space of a tensor using block circulant…
Using supercharacter theory, we identify the matrices that are diagonalized by the discrete cosine and discrete sine transforms, respectively. Our method affords a combinatorial interpretation for the matrix entries.
The computation of generalized inverses of quaternion matrices is a fundamental problem in quaternion linear algebra, with wide-ranging applications in signal processing, image restoration, and multidimensional data analysis. This paper…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…
Based on the definition of the Fourier transform in terms of the number operator of the quantum harmonic oscillator and in the corresponding definition of the fractional Fourier transform, we have obtained the discrete fractional Fourier…
By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion…
We present a practical and efficient means to compute the singular value decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to a real bidiagonal matrix B using quaternionic Householder transformations. Computation…
Dimensional regularization is incompatible with the standard covariant projection methods that are used to calculate the short-distance coefficients in inclusive heavy quarkonium production and annihilation rates. A new method is developed…
The cost of data input can dominate the run-time of quantum algorithms. Here, we consider data input of arithmetically structured matrices via block encoding circuits, the input model for the quantum singular value transform and related…
Transformations in the field of computer graphics and geometry are one of the most important concepts for efficient manipulation and control of objects in 2-dimensional and 3-dimensional space. Transformations take many forms each with…
Many tight frames of interest are constructed via their Gramian matrix (which determines the frame up to unitary equivalence). Given such a Gramian, it can be determined whether or not the tight frame is projective group frame, i.e., is the…