Improved tests and characterizations of totally nonnegative matrices

Abstract

Abstract. Totally nonnegative matrices, i.e., matrices having all minors nonnegative, are considered. A condensed form of the Cauchon algorithm which has been proposed for finding a parameterization of the set of these matrices with a fixed pattern of vanishing minors is derived. The close connection of this variant to Neville elimination and bidiagonalization is shown and new determinantal tests for total nonnegativity are developed which require much fewer minors to be checked than for the tests known so far. New characterizations of some subclasses of the totally nonnegative matrices as well as shorter proofs for some classes of matrices for being (nonsingular and) totally nonnegative are derived.

Further applications of the Cauchon algorithm to rank determination and bidiagonal factorization

“…In [3], [11] algorithms are presented which construct for a given Cauchon diagram C and any fixed square of C a lacunary sequence (with respect to C) starting at this square. The following proposition was given in [11,Proposition 4.1] with a sign condition which was removed in [4, Proposition 4.11].…”

“…We apply the Restoration algorithm (see [9] or [1]), which is the inverse procedure to the Cauchon algorithm, toG and obtain the matrix G . SinceG tends toG as tends to 0, it is easy to see that G tends to G. Hence A has a ESEB factorization which can be obtained from the elements ofG as in the case that A is nonsingular and totally nonnegative (see [1, p.52], [3,Section 4], although the analysis still applies in this case as well). Hence as tends to zero, we obtain the representation of l jk and u kj , k = 1, .…”

“…For properties of these matrices, the reader is referred to the monographs [7] or [12]. In [1], [3], a condensed form of the Cauchon algorithm was derived by which the amount of required arithmetic operations was reduced by one order of magnitude to bring it in line with the complexity of performing conventional Gaussian elimination, viz. O(n 3 ) for a nonsingular n-by-n matrix.…”

“…It is easy to see that ((q, j), The following example shows that the statement of Theorem 3.5 is not true if we allow the rows of A to be non-consecutive. If Theorem 3.5 would be valid also in the case that the rows to be checked for linear independence are non-consecutive, e.g., the first and the third row of A, we could apply Procedure 3.1 toÃ[1, 3|1, 2, 3, 4] which would result in a sequence γ containing only the pair (3,4). By Theorem 3.5 it would follow that both rows are linearly dependent, whereas the application of Procedure 3.1 to A[1, 3|1, 2, 3, 4] which is identical with the last two rows ofà gives the correct result.…”

Intervals of special sign regular matrices

“…As in [16], we relate to each entryã i 0 , j 0 ofà a sequence γ given by (12). It is sufficient to describe the construction of the sequence from the starting pair (i 0 , j 0 ) to the next pair (i 1 , j 1 ) with 0 <ã i 1 , j 1 (δ i 1 , j 1 < 0, see below) if i 1 , j 1 < n since for a given matrix A the determinantal test is performed by moving row by row from the bottom to the top row.…”