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• W2949283457 abstract "Let $X = (x_1,dots,x_n)$ be a vector of distinct positive integers. The $n times n$ matrix $S = S(X) := (gcd(x_i,x_j))_{i,j=1}^n$, where $gcd(x_i,x_j)$ denotes the greatest common divisor of $x_i$ and $x_j$, is called the greatest common divisor (GCD) matrix on $X$. By a surprising result of Beslin and Ligh [Linear Algebra and Appl. 118], all GCD matrices are positive definite. In this paper, we completely characterize the GCD matrices satisfying the stronger property of being totally nonnegative (TN) or totally positive (TP). As we show, a GCD matrix is never TP when $n geq 3$, and is TN if and only if it is $textrm{TN}_2$, i.e., all its $2 times 2$ minors are nonnegative. We next demonstrate that a GCD matrix is $textrm{TN}_2$ if and only if the exponents of each prime divisor in the prime factorization of the $x_i$s form a monotonic sequence. Reformulated in the language of kernels, our results characterize the subsets of integers over which the kernel $K(x,y) = gcd(x,y)$ is totally nonnegative. The proofs of our characterizations depend on Gantmacher and Krein's notion of a Green's matrix. We conclude by showing that a GCD matrix is TN if and only if it is a Green's matrix. As a consequence, we obtain explicit formulas for all the minors and for the inverse of totally nonnegative GCD matrices." @default.
• W2949283457 created "2019-06-27" @default.
• W2949283457 creator A5022321240 @default.
• W2949283457 creator A5077530909 @default.
• W2949283457 date "2019-01-07" @default.
• W2949283457 modified "2023-09-26" @default.
• W2949283457 title "Total nonnegativity of GCD matrices and kernels" @default.
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• W2949283457 doi "https://doi.org/10.48550/arxiv.1901.01947" @default.
• W2949283457 hasPublicationYear "2019" @default.
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