scholarly journals On the Kronecker Products and Their Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Huamin Zhang ◽  
Feng Ding
Keyword(s):  
Kronecker Product ◽  
Polynomial Matrix ◽  
Singular Values ◽  
Permutation Matrix ◽  
Kronecker Products ◽  
Mixed Matrix ◽  
Matrix Products ◽  
Vector Operator

This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. In addition, we establish the relations between the singular values of two matrices and their Kronecker product and the relations between the determinant, the trace, the rank, and the polynomial matrix of the Kronecker products.

scholarly journals On the Kronecker Product of Matrices and Their Applications To Linear Systems Via Modified QR-Algorithm

2021 ◽  
Vol 10 (6) ◽  
pp. 25352-25359
Author(s):  
Vellanki Lakshmi N. ◽  
Jajula Madhu ◽  
Musa Dileep Durani
Keyword(s):  
Kronecker Product ◽  
Polynomial Matrix ◽  
Singular Value ◽  
Least Square ◽  
Kronecker Products ◽  
Product System ◽  
Qr Algorithm ◽  
Value Decomposition ◽  
The Relationship ◽  
Product Of Matrices

This paper studies and supplements the proofs of the properties of the Kronecker Product of two matrices of different orders. We observe the relation between the singular value decomposition of the matrices and their Kronecker product and the relationship between the determinant, the trace, the rank and the polynomial matrix of the Kronecker products.  We also establish the best least square solutions of the Kronecker product system of equations by using modified QR-algorithm.

A matrix realignment method for recognizing entanglement

2003 ◽  
Vol 3 (3) ◽  
pp. 193-202
Author(s):  
K. Chen ◽  
L.-A. Wu
Keyword(s):  
Kronecker Product ◽  
Singular Values ◽  
Quantum Systems ◽  
Entangled States ◽  
Approximation Technique ◽  
Separable State ◽  
Simple Method ◽  
Bipartite Quantum Systems ◽  
The Matrix ◽  
Degree Of Entanglement

Motivated by the Kronecker product approximation technique, we have developed a very simple method to assess the inseparability of bipartite quantum systems, which is based on a realigned matrix constructed from the density matrix. For any separable state, the sum of the singular values of the matrix should be less than or equal to $1$. This condition provides a very simple, computable necessary criterion for separability, and shows powerful ability to identify most bound entangled states discussed in the literature. As a byproduct of the criterion, we give an estimate for the degree of entanglement of the quantum state.

scholarly journals Random matrix products: Universality and least singular values

2020 ◽  
Vol 48 (3) ◽  
pp. 1372-1410
Author(s):  
Phil Kopel ◽  
Sean O’Rourke ◽  
Van Vu
Keyword(s):  
Random Matrix ◽  
Singular Values ◽  
Random Matrix Products ◽  
Matrix Products

scholarly journals Character Polynomials, their q-Analogs and the Kronecker Product

10.37236/85 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
A. M. Garsia ◽  
A. Goupil
Keyword(s):  
Numerical Calculation ◽  
Kronecker Product ◽  
Hecke Algebra ◽  
Kronecker Products ◽  
Proper Role ◽  
Special Cases ◽  
Natural Counterpart ◽  
Kronecker Coefficients ◽  
Combinatorial Construction

The numerical calculation of character values as well as Kronecker coefficients can efficently be carried out by means of character polynomials. Yet these polynomials do not seem to have been given a proper role in present day literature or software. To show their remarkable simplicity we give here an "umbral" version and a recursive combinatorial construction. We also show that these polynomials have a natural counterpart in the standard Hecke algebra ${\cal H}_n(q\, )$. Their relation to Kronecker products is brought to the fore, as well as special cases and applications. This paper may also be used as a tutorial for working with character polynomials in the computation of Kronecker coefficients.

scholarly journals Stability of Kronecker Products of Irreducible Characters of the Symmetric Group

10.37236/1471 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Ernesto Vallejo
Keyword(s):  
Lower Bound ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Irreducible Characters ◽  
Long Time ◽  
The Stability

F. Murnaghan observed a long time ago that the computation of the decompositon of the Kronecker product $\chi^{(n-a, \lambda_2, \dots )}\otimes \chi^{(n-b, \mu_2, \dots)}$ of two irreducible characters of the symmetric group into irreducibles depends only on $\overline\lambda=(\lambda_2,\dots )$ and $\overline\mu =(\mu_2,\dots )$, but not on $n$. In this note we prove a similar result: given three partitions $\lambda$, $\mu$, $\nu$ of $n$ we obtain a lower bound on $n$, depending on $\overline\lambda$, $\overline\mu$, $\overline\nu$, for the stability of the multiplicity $c(\lambda,\mu,\nu)$ of $\chi^\nu$ in $\chi^\lambda \otimes \chi^\mu$. Our proof is purely combinatorial. It uses a description of the $c(\lambda,\mu,\nu)$'s in terms of signed special rim hook tabloids and Littlewood-Richardson multitableaux.

scholarly journals Multidimensional Arrays, Indices and Kronecker Products

Econometrics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 18
Author(s):  
D. Stephen G. Pollock
Keyword(s):  
Matrix Algebra ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Multidimensional Arrays ◽  
Conventional Notation ◽  
Index Notation ◽  
Product Of Matrices

Much of the algebra that is associated with the Kronecker product of matrices has been rendered in the conventional notation of matrix algebra, which conceals the essential structures of the objects of the analysis. This makes it difficult to establish even the most salient of the results. The problems can be greatly alleviated by adopting an orderly index notation that reveals these structures. This claim is demonstrated by considering a problem that several authors have already addressed without producing a widely accepted solution.

Face Recognition Using 2D-FLDA Based on Approximate SVD Obtained with Kronecker Products with One Training Sample

2019 ◽  
pp. 1-9
Author(s):  
Ümit Çiğdem Turhal
Keyword(s):  
Face Recognition ◽  
Kronecker Product ◽  
Dimensional Space ◽  
Recognition Performance ◽  
Recognition Task ◽  
Linear Structure ◽  
Training Sample ◽  
Kronecker Products ◽  
Linear Discriminant ◽  
Lower Dimensional

Aims: In a face recognition task, it is a challenging problem to find lots of images for a person. Even, sometimes there can be only one image, available for a person. In these cases many of the methods are exposed to serious performance drops even some of these fail to work. Recently this problem has become remarkable for researchers. In some of these studies the database is extended using a synthesized image which is constructed from the singular value decomposition (SVD) of the single training image. In this paper, for such a method, SVD based 2 Dimensional Fisher Linear Discriminant Analysis (2D-FLDA), it is proposed a new approach to find the SVD of the image matrix with the aim of to increase the recognition performance. Study Design: In this paper, in a face recognition task with 2D-FLDA, in one training sample case, instead of original SVD of the image matrix, the approximate SVD of its based on multiple kronecker product sums is used. In order to obtain it, image matrix is first reshaped thus it is to be lower dimensional matrices and, then the sum of multiple kronecker products (MKPS) is applied in this lower dimensional space. Methodology: Experiments are performed on two known databases Ar-Face and ORL face databases. The performance of the proposed method is evaluated when there are facial expression, lightning conditions and pose variations. Results: In each experiment, the approximate SVD approach based on multiple kronecker product sum gets approximately 3% better results when compared with the original SVD. Conclusion: Experimental results verify that the proposed method achieves better recognition performance over the traditional one. The reason for this is the proposed approximate SVD has the advantages of simplicity, and also as the kronecker factors possess additional linear structure, kronecker product can capture potential self-similarity.

Kronecker Products and Local Joins of Graphs

1977 ◽  
Vol 29 (2) ◽  
pp. 255-269 ◽  
Author(s):  
M. Farzan ◽  
D. A. Waller
Keyword(s):  
Tensor Product ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Finite Graphs ◽  
Product Of Graphs

When studying the category raph of finite graphs and their morphisms, Ave can exploit the fact that this category has products, [we define these ideas in detail in § 2]. This categorical product of graphs is usually called their Kronecker product, though it has been approached by various authors in various ways and under various names, including tensor product, cardinal product, conjunction and of course categorical product (see for example [6; 7; 11 ; 14; 17 and 23]).

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