scholarly journals ON THE REAL MATRIX REPRESENTATION OF PT-SYMMETRIC OPERATORS

2014 ◽  
Vol 54 (2) ◽  
pp. 113-115 ◽  
Author(s):  
Francisco M. Fernández

We discuss the construction of real matrix representations of PT-symmetric operators. We show the limitation of a general recipe presented some time ago for non-Hermitian Hamiltonians with antiunitary symmetry and propose a way to overcome it. Our results agree with earlier ones for a particular case.

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 16
Author(s):  
Rogério Serôdio ◽  
Patricia Beites ◽  
José Vitória

A pseudo real matrix representation of an octonion, which is based on two real matrix representations of a quaternion, is considered. We study how some operations defined on the octonions change the set of eigenvalues of the matrix obtained if these operations are performed after or before the matrix representation. The established results could be of particular interest to researchers working on estimation algorithms involving such operations.


2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Cennet Bolat ◽  
Ahmet İpek

Some complex quaternionic equations in the typeAX-XB=Care investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.


2020 ◽  
Vol 34 (05) ◽  
pp. 9330-9337
Author(s):  
Dong Xu ◽  
Wu-Jun Li

Answer selection is an important subtask of question answering (QA), in which deep models usually achieve better performance than non-deep models. Most deep models adopt question-answer interaction mechanisms, such as attention, to get vector representations for answers. When these interaction based deep models are deployed for online prediction, the representations of all answers need to be recalculated for each question. This procedure is time-consuming for deep models with complex encoders like BERT which usually have better accuracy than simple encoders. One possible solution is to store the matrix representation (encoder output) of each answer in memory to avoid recalculation. But this will bring large memory cost. In this paper, we propose a novel method, called hashing based answer selection (HAS), to tackle this problem. HAS adopts a hashing strategy to learn a binary matrix representation for each answer, which can dramatically reduce the memory cost for storing the matrix representations of answers. Hence, HAS can adopt complex encoders like BERT in the model, but the online prediction of HAS is still fast with a low memory cost. Experimental results on three popular answer selection datasets show that HAS can outperform existing models to achieve state-of-the-art performance.


2009 ◽  
Vol 2009 ◽  
pp. 1-5
Author(s):  
Marina Arav ◽  
Frank Hall ◽  
Zhongshan Li ◽  
Bhaskara Rao

Let H be an m×n real matrix and let Zi be the set of column indices of the zero entries of row i of H. Then the conditions |Zk∩(∪i=1k−1Zi)|≤1 for all k  (2≤k≤m) are called the (row) Zero Position Conditions (ZPCs). If H satisfies the ZPC, then H is said to be a (row) ZPC matrix. If HT satisfies the ZPC, then H is said to be a column ZPC matrix. The real matrix H is said to have a zero cycle if H has a sequence of at least four zero entries of the form hi1j1,hi1j2,hi2j2,hi2j3,…,hikjk,hikj1 in which the consecutive entries alternatively share the same row or column index (but not both), and the last entry has one common index with the first entry. Several connections between the ZPC and the nonexistence of zero cycles are established. In particular, it is proved that a matrix H has no zero cycle if and only if there are permutation matrices P and Q such that PHQ is a row ZPC matrix and a column ZPC matrix.


1997 ◽  
Vol 12 (21) ◽  
pp. 1573-1582 ◽  
Author(s):  
Alfredo Herrera-Aguilar ◽  
Oleg Kechkin

A Ernst-like matrix representation of (3+d)-dimensional Einstein–Kalb–Ramond theory is developed. The analogy with the Einstein and Einstein–Maxwell–Dilaton–Axion theories is discussed. The subsequent reduction to two dimensions is considered. It is shown that, in this case, the theory allows two different Ernst-like d×d-matrix formulations: the real nondualized target space, and the Hermitian dualized nontarget space one. The O(d, d)-symmetry is written in an SL (2,R) matrix-valued form in both cases. The Kramer–Neugebauer transformation, which algebraically maps the nondualized Ernst potential onto the dualized one, is presented.


2019 ◽  
Vol 11 (1) ◽  
pp. 1-6
Author(s):  
Mil Mascaras ◽  
Jeffrey Uhlmann

In this paper we derive a representation of an arbitrary real matrix M as the difference of a real matrix A and the transpose of its inverse. This expression may prove useful for progressing beyond known results for which the appearance of transpose-inverse terms prove to be obstacles, particularly in control theory and related applications such as computational simulation and analysis of matrix representations of articulated figures.


2017 ◽  
Vol 101 (10) ◽  
pp. 2333-2347
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Junesang Choi ◽  
Shahid Mubeen ◽  
Muhammad Arshad
Keyword(s):  

2015 ◽  
Vol 08 (03) ◽  
pp. 1550042 ◽  
Author(s):  
Junying Guo ◽  
Xiaojiang Guo ◽  
K. P. Shum

The properties of right ample semigroups have been extensively considered and studied by many authors. In this paper, we concentrate on the matrix representations of right ample semigroups. The (left; right) uniform matrix representation is initially defined. After some properties of left uniform matrix representations of a right ample semigroup are given, we prove that any irreducible left uniform representations of a right ample semigroup can be obtained by using an irreducible left uniform representation of some primitive right ample semigroup. In particular, a construction theorem of prime left uniform representation of right ample semigroups is established.


2001 ◽  
Vol 16 (10) ◽  
pp. 647-654
Author(s):  
CHANDRARAJU CVAVB

Fritzsch type of real symmetric 3×3 matrices are chosen. From these matrices the mass matrices relevant to the four-quark cases are deduced. It is also shown that the orthogonal matrices that diagonalize the 3×3 Fritzsch mass matrices also yield the orthogonal matrices that diagonalize the mass matrices corresponding to the four-quark case. The Cabibbo mixing matrix is straightaway obtained from the KM real mixing matrix. An exact expression for the Cabibbo mixing angle is found here. The real KM matrix is reduced to a few parameters, which can be determined from the experiment. The results obtained here are the exact expressions for the KM real matrix.


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