Copyright © 2016 Mehsin Jabel Atteya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. MJ Journal on Algebra and Its Applications 1 (1) (2016) 1-4 Website: www.math-journals.com/index.php/JAA doi: 10.14419/jaa.v1i1.35 Research paper Kronecker product with applications Mehsin Jabel Atteya * Al-Mustansiriyah University, College of Education, Department of Mathematics, IRAQ *Corresponding author E-mail: mehsinatteya@yahoo.com Abstract The main purpose of this paper study the property of the Kronecker product related to the Kronecker’s Delta and determi- nants of matrices .This product gives the possibility to obtain a commutativity of any pair of square matrices such as Hankel matrices and strictly upper triangular matrices. Kronecker product works without the assumptions on the size of composing matrices. Keywords: Kronecker Products, Kronecker’s Delta, Square Matrices, Hankel Matrices, Strictly Upper Triangular Matrices, Commut a- tive Matrices. AMS Subject Classification: 15A09, 15A27, 15A69 1. Introduction 'On the History of the Kronecker Product' this title of the paper which wrote by Henderson, Pukelsheim, and Searle [1].Apparently, the first documented work on Kronecker products was written by Johann Georg Zehfuss between 1858 and 1868. In fact, Zehfuss who found the determinant result of a two square matrices A and B with their dimension x and y respectively |A ⊗ B| = |A| y |B| x , (*).In Berlin, Kronecker gave a series of lectures in the 1880’s, where he introduced the result (*) to his students. Later, in the 1890’s, Hurwitz and St´ephanos developed the same determinant equality and other results involving Kronecker products such as: (A ⊗ B) (C ⊗ D) = (AC) ⊗ (BD), In addition to that, Hurwitz used the symbol × to denote the operation. Furthermore, St´ephanos derives the result that the eigenvalues of A⊗B are the products of all eigenvalues of A with all eigenvalues of B. Rados is one of many writers work in this field in the late 1800’s who also discovered property (*) independently. Rados even thought that he wrote the original paper on property (*) and claims it for himself in his paper published in 1900, questioning Hensel’s contributing it to Kronecker. In spite of Rados’ claim, the determinant result (*) continued to be associated with Kronecker. In the 1930’s, even the definition of the matrix oper a- tion A⊗B was associated with Kronecker’s name. Today, we know the Kronecker product as “Kronecker” product and not as “Zehfuss”, “Hurwitz”, “St´ephanos”, or “Rados” product. Huamin Zhang and Feng Ding [2] proved, if A ∈ F m×n and B ∈ F p×q , then one has P݌(A⊗B)P   ݍ= B ⊗ A. The Kronecker product has an important role in the linear matrix equation theory. The solution of the Sylvester and the Sylvester-like equations is a hotspot research area. Recently, the innovational and computationally efficient numerical algorithms based on the hierarchical identification principle for the generalized Sylvester matrix equations [3,4] and coupled matrix equations [5, 6] were proposed by Ding and Chen. On the other hand, the iterative algorithms for the extended Sylvester-conjugate matrix equations were discussed in [7], [8]. Other related work is included in [9–11]. Charles F. Van Loan [12] point to the widening use of the Kronecker product in numerical linear al- gebra. Amy N. Langville and William J.Stewart [13] proved the condition number: For all matrix norms, cond (A ⊗ B) = cond(A) cond(B). Recently, Yong Sik Yun and Chul Kang [14] derived further properties and results of Kronecker prod- ucts, vec-operator and commutation matrices. This paper will look at applications of the Kronecker's Delta, determinant property with Kronecker product in a square matrix.