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Determinants in the Kronecker product of matrices: the incidence matrix of a complete graph. (English) Zbl 1218.15003

Summary: We investigate the least common multiple of all subdeterminants, lcmd\((A \otimes B)\), of a Kronecker product of matrices, of which one is an integral matrix \(A\) with two columns and the other is the incidence matrix of a complete graph with \(n\) vertices. We prove that this quantity is the least common multiple of lcmd\((A)\) to the power \(n - 1\) and certain binomial functions of the entries of \(A\).

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
15B36 Matrices of integers
15A69 Multilinear algebra, tensor calculus
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)

References:

[1] Chaiken S, A q-queens problem
[2] Horn RA, Topics in Matrix Analysis (1991)
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