Let A be a real n × n matrix with non-negative non-diagonal elements ai j (i ≠ J). A is called a stability matrix if and only if all of its eigenvalues have strictly negative real parts. It is proved that A is a stability matrix if and only if the system Ax ≽ 0, x ≽ 0 has no non-trivial solution. Further, one and only one of the systems Ax < 0, x > 0 and Ax ≽ 0, x ≽ 0 has a non-trivial solution.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory