The Weak Commutation Matrices of Matrix with Duplicate Entries in Its Main Diagonal

Yanita Yanita, Nia Rili Putri Irawan, Lyra Yulianty, Nova Noliza Bakar


This article discusses the permutation matrix which is a weak commutation matrix. This weak commutation matrix is determined by compiling the duplicate entry in the main diagonal, i.e., two, three, and four which are the same in different positions. The method for determining this matrix is to use the property of  the transformation  matrix to  transpose the matrix.  Based on this, we have 24 weak commutation matrices for the matrices.


Weak Commutation Matrix; Duplicate Entries; Diagonal Entry; Vec Matrix

Full Text:



D. A. Harville, Matrix Algebra from a Statistician's Perspective, New York: Springer, 2008.

J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, USA: John Wiley and Sons, Inc, 2019.

J. R. Magnus and H. Neudecker, "The commutation matrix: some properties and application," The Annals of Statistics, vol. 7, no. 2, pp. 381-394, 1979.

C. Xu, H. Lingling and L. Zerong, "Commutation matrices and commutation tensor," Linear and Multilinear Algebra, vol. 68, no. 9, pp. 1721-1742, 2020.

J. A. Galian, Contemporary Abstract Algebra, 7 ed., Belmon, CA: Brooks/Cole, Cengage Learning, 2010.

R. Piziak and P. L. Odell, Matrix Theory: From Generalized Inverses to Jordan Form, New York: Chapmann & Hall/CRC, 2007.

H. Zhang and F. Ding, "On the Kronecker products and their applications," Journal of Applied Mathematics, pp. 1-8, 2013.

Y. Yanita, E. Purwanti and L. Yulianti, "The commutation matrices of elements in Kronecker quaternion group," Jambura Journal of mathematics, vol. 4, no. 1, pp. 135-144, 2022.

Y. Yanita, A. M. Zakiya dan M. R. Helmi, “Solvability group from Kronecker product on the representation of quaternion group,” Asian Journal of Scientifics Research, pp. 293-297, 2018.



  • There are currently no refbacks.

Copyright (c) 2023 Yanita Yanita, Nia Rili Putri Irawan, Lyra Yulianty, Nova Noliza Bakar

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.