On Cartesian product of Euclidean distance matrices
| Autor: | Hiroshi Kurata, Ravindra B. Bapat |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Kronecker product
Numerical Analysis Algebra and Number Theory 010102 general mathematics Minor (linear algebra) 010103 numerical & computational mathematics Function (mathematics) Rank (differential topology) Cartesian product 01 natural sciences Combinatorics symbols.namesake Matrix (mathematics) Product (mathematics) symbols Discrete Mathematics and Combinatorics Order (group theory) Geometry and Topology 0101 mathematics Mathematics |
| Zdroj: | Linear Algebra and its Applications. 562:135-153 |
| ISSN: | 0024-3795 |
| Popis: | If A ∈ R m × m and B ∈ R n × n , we define the product A ⊘ B as A ⊘ B = A ⊗ J n + J m ⊗ B , where ⊗ denotes the Kronecker product and J n is the n × n matrix of all ones. We refer to this product as the Cartesian product of A and B since if D 1 and D 2 are the distance matrices of graphs G 1 and G 2 respectively, then D 1 ⊘ D 2 is the distance matrix of the Cartesian product G 1 □ G 2 . We study Cartesian products of Euclidean distance matrices (EDMs). We prove that if A and B are EDMs, then so is the product A ⊘ B . We show that if A is an EDM and U is symmetric, then A ⊗ U is an EDM if and only if U = c J n for some c. It is shown that for EDMs A and B, A ⊘ B is spherical if and only if both A and B are spherical. If A and B are EDMs, then we derive expressions for the rank and the Moore–Penrose inverse of A ⊘ B . In the final section we consider the product A ⊘ B for arbitrary matrices. For A ∈ R m × m , B ∈ R n × n , we show that all nonzero minors of A ⊘ B of order m + n − 1 are equal. An explicit formula for a nonzero minor of order m + n − 1 is proved. The result is shown to generalize the familiar fact that the determinant of the distance matrix of a tree on n vertices does not depend on the tree and is a function only of n. |
| Databáze: | OpenAIRE |
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