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**Moore-Penrose inverse positivity of interval matrices.**
*(English)*
Zbl 1236.15012

The authors study (entrywise) nonnegativity of the Moore-Penrose inverse \(C^†\) of a matrix \(C\in\mathbb{R}^{m\times n}\). In particular they consider the following questions:

(1) When is \(C^†\) nonnegative? (2) When is \(C^†\) nonnegative if \(C\leq B\), where \(B\in \mathbb{R}^{m\times n}\) is a given matrix with \(B^†\geq 0\)? (3) When is \(C^†\) nonnegative for all matrices \(C\in[A, B]\), where \(A^†\geq 0\) and \(B^†\geq 0\)?

Each of these three questions is answered by necessary and sufficient conditions which are similar to known ones for regular \(n\times n\) matrices \(A\), \(B\), \(C\) in the same connection. One of these known criteria comes from Ortega and Rheinboldt and uses weak regular splittings, another one goes back to Krasnosel’skij, Lifshits, and Sobolev and uses some part of the range of \(C\), and a third one was published by Rohn and answers question (3) for regular matrices.

For the analogue of the latter one the concept of a range kernel regular interval matrix \([A,B]\) is introduced by \(R(A)= R(B)\) and \(N(A)= N(B)\), where \(R(\cdot)\) denotes the range and \(N(\cdot)\) the kernel of an \(m\times n\) matrix. For such an interval matrix a particular subset \(K\) is introduced by virtue of which various additional results could be derived.

(1) When is \(C^†\) nonnegative? (2) When is \(C^†\) nonnegative if \(C\leq B\), where \(B\in \mathbb{R}^{m\times n}\) is a given matrix with \(B^†\geq 0\)? (3) When is \(C^†\) nonnegative for all matrices \(C\in[A, B]\), where \(A^†\geq 0\) and \(B^†\geq 0\)?

Each of these three questions is answered by necessary and sufficient conditions which are similar to known ones for regular \(n\times n\) matrices \(A\), \(B\), \(C\) in the same connection. One of these known criteria comes from Ortega and Rheinboldt and uses weak regular splittings, another one goes back to Krasnosel’skij, Lifshits, and Sobolev and uses some part of the range of \(C\), and a third one was published by Rohn and answers question (3) for regular matrices.

For the analogue of the latter one the concept of a range kernel regular interval matrix \([A,B]\) is introduced by \(R(A)= R(B)\) and \(N(A)= N(B)\), where \(R(\cdot)\) denotes the range and \(N(\cdot)\) the kernel of an \(m\times n\) matrix. For such an interval matrix a particular subset \(K\) is introduced by virtue of which various additional results could be derived.

Reviewer: Günter Mayer (Rostock)

### Keywords:

interval matrix; \(M\)-matrix; Moore-Penrose inverse; range kernel regularity; proper splitting; weak regular splitting; inverse positive matrix
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\textit{M. Rajesh Kannan} and \textit{K. C. Sivakumar}, Linear Algebra Appl. 436, No. 3, 571--578 (2012; Zbl 1236.15012)

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### References:

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