A theorem of the alternatives for the equation<i>Ax</i>+<i>B</i>|<i>x</i>| =<i>b</i> 论文

摘要

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.