On matrices with different tropical and Kapranov ranks
Ya. N. Shitov · Mathematical Notes · 2012
The main object of tropical mathematics is the tropical semiring, i.e., the set R of real numbers with the operations of tropical addition, ⊕, and tropical multiplication, ⊗, which are defined by the rules a⊕ b = min{a, b} and a⊗ b = a+ b for any a, b ∈ R. Note that the element 0 ∈ R is neutral with respect to the tropical multiplication. Tropical mathematics is obtained from traditional mathematics as a result of Maslov dequantization (see [1]–[3]). Methods of tropical mathematics turn out to be useful for numerous applications (see [4]–[6]) and are employed in algebraic geometry (see [7], [8]). The notion of rank of tropical matrices is of great interest (see [4], [9], [10]). In contrast to the case of matrices over fields, there is a lot of different rank functions for tropical matrices, and many of them are described in [4], [9], and [10]. The present note is devoted to the notions of tropical rank and Kapranov rank. We use the symbol F to denote a field and F∗ for the set of nonzero elements of F. Denote by aij the entries of a matrix A, by A(j) the jth column and by A(i) the ith row of the matrix, and by A the matrix transposed to A. A submatrix of a matrix A formed by the rows with indices r1, . . . , rk is denoted by A[r1, . . . , rk] and a submatrix of the matrix A[r1, . . . , rk] formed by the columns with indices c1, . . . , cl is denoted by A[r1, . . . , rk|c1, . . . , cl]. Denote byHF the field (see [11]) consisting of formal sums of the form