On Cartesian product of matrices
DOI :
https://doi.org/10.5269/bspm.63383Résumé
Bapat and Kurata [\textit{Linear Algebra Appl.}, 562(2019), 135-153] defined the Cartesian product of two square matrices $A$ and $B$ as $A\oslash B=A\otimes \J+\J\otimes B$, where $\J$ is the matrix of all one of appropriate order and $\otimes$ is the Kronecker product. In this article, we find the expression for the trace of the Cartesian product of any finite number of square matrices in terms of the traces of the individual matrices. Also, we establish some identities involving the Cartesian product of matrices. Finally, we apply the Cartesian product to study some graph-theoretic properties.
Références
2. Bapat, R. B. and Kurata, H. On Cartesian product of Euclidean distance. Linear Algebra and its Applications, 526,135-153, (2019).
3. Graham, A. Kronecker Products and matrix Calculus: with applications. John wiley & Sons, New York, 1981.
4. Graham, R. L. and Pollak, H. O. On the addressing problem for loop switching. The Bell System Technical Journal, 50(8),2495-2519, (1971).
5. Zhang, F. Matrix Theory: Basic Results and Techniques. Springer, India, (2010).
6. Zhang, X. and Godsil, C. The inertia of distance matrices of some graphs. Discrete Mathematics, 313(16), 1655-1664, (2013).
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