scholarly journals On Hadamard Product of Hypercomplex Numbers

2021 ◽  
Vol 104 (4) ◽  
pp. 68-73
Author(s):  
A. Da¸sdemir ◽  
Keyword(s):  
Hadamard Product ◽  
Matrix Theory ◽  
Real Matrix ◽  
Complex Matrix ◽  
Hypercomplex Numbers ◽  
Matrix Space ◽  
The Real ◽  
Algebraic Properties ◽  
Product Rules ◽  
Analogous Product

Certain product rules take various forms in the set of hypercomplex numbers. In this paper, we introduce a new multiplication form of the hypercomplex numbers that will be called «the Hadamard product», inspired by the analogous product in the real matrix space, and investigate some algebraic properties of that, including the norm of inequality. In particular, we extend our new definition and its applications to the complex matrix theory.

scholarly journals Pseudo-hermitian random matrix theory: a review

2021 ◽  
Vol 2038 (1) ◽  
pp. 012009
Author(s):  
Joshua Feinberg ◽  
Roman Riser
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Complex Conjugate ◽  
Indefinite Metric ◽  
The Real ◽  
New Type ◽  
Diagrammatic Method

Abstract We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the complex plane and on the real axis, in the large-N, planar limit. In one of the models we discuss, the metric B depends on a certain real parameter t. As t varies, the model exhibits various ‘phase transitions’ associated with eigenvalues flowing from the complex plane onto the real axis, causing disjoint eigenvalue support intervals to merge. Our analytical results agree well with presented numerical simulations.

scholarly journals ZPC Matrices and Zero Cycles

2009 ◽  
Vol 2009 ◽  
pp. 1-5
Author(s):  
Marina Arav ◽  
Frank Hall ◽  
Zhongshan Li ◽  
Bhaskara Rao
Keyword(s):  
Real Matrix ◽  
The Real ◽  
Permutation Matrices ◽  
Zero Position ◽  
Common Index

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.

Appendix: Real and complex canonical forms

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Canonical Forms ◽  
Real Matrix ◽  
Complex Matrix ◽  
Complex Matrices ◽  
Matrix Pencils ◽  
Real Matrices

This chapter presents canonical forms for real and complex matrices and for pairs of real and complex matrices, or matrix pencils, with symmetries. All these forms are known, and most are well-known. The chapter first looks at Jordan and Kronecker canonical forms, before turning to real matrix pencils with symmetries. It provides canonical forms for pairs of real matrices, either one of which is symmetric or skewsymmetric, or what is the same, corresponding matrix pencils. Finally, this chapter presents canonical forms of complex matrix pencils with various symmetries, such as complex matrix pencils with symmetries with respect to transposition.

EXTENSIONS OF THE REAL MATRIX-VARIATE GAMMA AND BETA FUNCTIONS AND THEIR APPLICATIONS

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

REAL KM MATRIX AND CABIBBO MATRIX

2001 ◽  
Vol 16 (10) ◽  
pp. 647-654
Author(s):  
CHANDRARAJU CVAVB
Keyword(s):  
Exact Expression ◽  
Real Matrix ◽  
Orthogonal Matrices ◽  
Mixing Angle ◽  
The Real ◽  
Mass Matrices ◽  
Mixing Matrix

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.

Permutations preserving sums of rearranged real series

2013 ◽  
Vol 11 (5) ◽  
Author(s):  
Roman Wituła
Keyword(s):  
The Real ◽  
Combinatorial Description ◽  
The Family ◽  
Algebraic Properties

AbstractThe aim of this paper is to discuss one of the most interesting and unsolved problems of the real series theory: rearrangements that preserve sums of series. Certain hypothesis about combinatorial description of the corresponding permutations is presented and basic algebraic properties of the family $\mathfrak{S}_0 $, introduced by it, are investigated.

scholarly journals Khatri-Rao Products for Operator Matrices Acting on the Direct Sum of Hilbert Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Hadamard Product ◽  
Operator Matrices ◽  
Linear Map ◽  
Direct Sum ◽  
Positive Linear Map ◽  
Algebraic Properties ◽  
Product Of Matrices

We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry.

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