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Jordan structures of irreducible totally nonnegative matrices with a prescribed sequence of the first p-indices. (English) Zbl 1486.15017

Summary: Let \(A \in\mathbb{R}^{n \times n}\) be an irreducible totally nonnegative matrix with rank \(r\) and principal rank \(p\), that is, \(A\) is irreducible with all minors nonnegative, \(r\) is the size of the largest invertible square submatrix of \(A\) and \(p\) is the size of its largest invertible principal submatrix. We consider the sequence \(\{1, i_2, \dots, i_p\}\) of the first \(p\)-indices of \(A\) as the first initial row and column indices of a \(p \times p\) invertible principal submatrix of \(A\). A triple \((n, r, p)\) is called \((1, i_2, \dots, i_p)\)-realizable if there exists an irreducible totally nonnegative matrix \(A \in\mathbb{R}^{n \times n}\) with rank \(r\), principal rank \(p\), and \(\{1, i_2, \dots, i_p\}\) is the sequence of its first \(p\)-indices. In this work we study the Jordan structures corresponding to the zero eigenvalue of irreducible totally nonnegative matrices associated with a triple \((n, r, p)\) \((1, i_2, \dots, i_p)\)-realizable.

MSC:

15A20 Diagonalization, Jordan forms
15A21 Canonical forms, reductions, classification
15A03 Vector spaces, linear dependence, rank, lineability
15A29 Inverse problems in linear algebra
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