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On the Cayley transform of positivity classes of matrices. (English) Zbl 1023.15009

The Cayley transform of \(A\), \(F=(I+A)^{-1} (I-A)\), is studied when \(A\) is a \(P\)-matrix, an \(M\)-matrix, an inverse \(M\)-matrix, a positive definite matrix, or a totally nonnegative matrix. Given a matrix \(A\) in each of these positivity classes and using the fact that the Cayley transform is an involution, properties of the ensuing factorization \(A=(I+F)^{-1} (I-F)\) are examined. Specifically, it is investigated whether these factors belong to the same positivity class as \(A\) and, conversely, under what conditions on these factors does \(A\) belong to one of the above positivity classes.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A23 Factorization of matrices
15A24 Matrix equations and identities
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