Generalized totally positive matrices

✍ Scribed by Miroslav Fiedler; Thomas L. Markham

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 111 KB
• Volume: 306
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00240-2

✦ Synopsis

We say that a rectangular matrix over a (in general, noncommutative) ring with identity having a positive part is generalized totally positive (GTP) if in all nested sequences of socalled relevant submatrices, the Schur complements are positive. Here, a relevant submatrix is such either having k consecutive rows and the first k columns, or k consecutive columns and the first k rows. This notion generalizes the usual totally positive matrices. We prove e.g. that a square matrix is GTP if and only if it admits a certain factorization with bidiagonaltype factors and certain invertible entries. Also, the product of square GTP-matrices of the same order is again a GTP-matrix, and its inverse has the checkerboard-sign property.