A theorem of the alternatives for the equationAx+B|x| =b
Top Cited Papers
- 1 November 2004
- journal article
- research article
- Published by Taylor & Francis in Linear and Multilinear Algebra
- Vol. 52 (6), 421-426
- https://doi.org/10.1080/0308108042000220686
Abstract
The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B|x| = b has a unique solution for each B with |B| ≤ D and for each b, or the equation Ax + B 0|x| = 0 has a nontrivial solution for some matrix B 0 of a very special form, |B 0| ≤ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax| = λ|x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue.Keywords
This publication has 6 references indexed in Scilit:
- Checking robust nonsingularity is NP-hardMathematics of Control, Signals, and Systems, 1993
- Interval Matrices: Singularity and Real EigenvaluesSIAM Journal on Matrix Analysis and Applications, 1993
- Systems of linear interval equationsLinear Algebra and its Applications, 1989
- On the number of solutions to the complementarity problem and spanning properties of complementary conesLinear Algebra and its Applications, 1972
- Zur Theorie der MatricesMathematische Annalen, 1907
- Theorie der einfachen Ungleichungen.Journal für die reine und angewandte Mathematik (Crelles Journal), 1902