Orthogonal bases that lead to symmetric nonnegative matrices

Elsner L, Nabben R, Neumann M (1998)
Linear Algebra and its Applications 271: 323-343.

Zeitschriftenaufsatz | Veröffentlicht| Englisch
 
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Autor/in
Elsner, LudwigUniBi; Nabben, Reinhard; Neumann, Michael
Abstract / Bemerkung
In a paper dating back to 1983, Soules constructs from a positive vector x an orthogonal matrix R which has the property that for any nonnegative diagonal matrix Lambda with nonincreasing diagonal entries, the matrix R Lambda R-T has all its entries nonnegative. Independently, Fiedler in 1988 showed that any symmetric irreducible nonsingular matrix whose powers are all M-matrices (and hence an MMA-matrix in the language of Friedland, Hershkowitz, and Schneider) must have an orthogonal matrix of eigenvectors (R) over tilde which has similar properties to those of R. Here, for a given positive n-vector x, we investigate the structure of all orthogonal matrices R for which, for any nonnegative diagonal matrix Lambda as above, the matrices R Lambda R-T are nonnegative. Up to a permutation of its columns, each such R corresponds to a binary tree whose vertices are subsets of the set {1, 2,..., n} with the property that each vertex has either no successor or exactly two disjoint successors. For such orthogonal matrices R and such nonsingular diagonal matrices Lambda, we show that the set of matrices of the form R Lambda R-T and the set of inverse MMA-matrices (i.e. matrices whose inverses are MMA-matrices) coincide. Using this result, we establish a relation between strictly ultrametric matrices and inverse MMA-matrices. Finally, we show that the QR factorization of R Lambda R-T, for certain such R's, has a special sign pattern. (C) 1998 Elsevier Science Inc.
Erscheinungsjahr
1998
Zeitschriftentitel
Linear Algebra and its Applications
Band
271
Seite(n)
323-343
ISSN
0024-3795
Page URI
https://pub.uni-bielefeld.de/record/1641961

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Elsner L, Nabben R, Neumann M. Orthogonal bases that lead to symmetric nonnegative matrices. Linear Algebra and its Applications. 1998;271:323-343.
Elsner, L., Nabben, R., & Neumann, M. (1998). Orthogonal bases that lead to symmetric nonnegative matrices. Linear Algebra and its Applications, 271, 323-343.
Elsner, L., Nabben, R., and Neumann, M. (1998). Orthogonal bases that lead to symmetric nonnegative matrices. Linear Algebra and its Applications 271, 323-343.
Elsner, L., Nabben, R., & Neumann, M., 1998. Orthogonal bases that lead to symmetric nonnegative matrices. Linear Algebra and its Applications, 271, p 323-343.
L. Elsner, R. Nabben, and M. Neumann, “Orthogonal bases that lead to symmetric nonnegative matrices”, Linear Algebra and its Applications, vol. 271, 1998, pp. 323-343.
Elsner, L., Nabben, R., Neumann, M.: Orthogonal bases that lead to symmetric nonnegative matrices. Linear Algebra and its Applications. 271, 323-343 (1998).
Elsner, Ludwig, Nabben, Reinhard, and Neumann, Michael. “Orthogonal bases that lead to symmetric nonnegative matrices”. Linear Algebra and its Applications 271 (1998): 323-343.