Totally positive matrices

A stable test for strict sign regularity

“…Since A is SSR, when we apply Neville elimination with two-determinant pivoting, the resulting matrixà (1) is SSR with ε 2 (à (1) ) = +1. Let us consider α 1 , β 1 with 2 ≤ α 1 ≤ m and 2 ≤ β 1 ≤ n. Since ε 2 (à (1) ) = +1, applying formula (2.4) (taking C :=à (1) [α 1 − 1, α 1 |1, β 1 ] and γ := (1)), we can deduce that all the elements of the matrix A (2) [2, .…”

“…Let us consider α 1 , β 1 with 2 ≤ α 1 ≤ m and 2 ≤ β 1 ≤ n. Since ε 2 (Ã (1) ) = +1, applying formula (2.4) (taking C :=Ã (1) [α 1 − 1, α 1 |1, β 1 ] and γ := (1)), we can deduce that all the elements of the matrix A (2) [2, . .…”

“…. , n] have the same strict sign which is ε s+1 (Ã (1) )/ε 1 (Ã (1) (A)). From this last fact, and taking into account that the first row of A (k) coincides with either the first or the last row of A, we conclude that ε 1 (A (k) ) = ε 1 (A) for all k.…”

“…The interest in these matrices comes from their characterizations as variation-diminishing linear maps: the number of sign changes in the consecutive components of the image of a vector is bounded above by the number of sign changes in the consecutive components of the vector (cf. Theorem 5.6 of [1]). The theory of variation-diminishing transformations was originated by Schoenberg [16].…”

“…The theory of variation-diminishing transformations was originated by Schoenberg [16]. Many applications of these transformations can be found in [9] and [1]. Let us now mention some examples showing the usefulness of providing efficient tests to check if a given matrix is SSR.…”