Multiple Models Approach in Automation: Takagi-Sugeno Fuzzy Systems

Appendix 2

Properties of M-matrices

These definitions can be found in [FOS 93].

DEFINITION A2.1.– Z-matrix

A matrix for which all off-diagonal terms are negative or null is called a Z-matrix.

THEOREM A2.1.– M-matrix

In order for a Z-matrix A to be an M-matrix, it is necessary and sufficient that one of the following equivalent properties is verified:

– each eigenvalue of A is of a positive real part;

– all of the principal minor determinants of A are positive;

– A⁻¹ exists and is nonnegative (all of its elements are either positive or null);

– there exists a positive vector υ (υ ≠ 0), such that the vector Aυ is also a strictly positive vector;

– A is a positive quasi-dominant diagonal matrix, i.e. there exists a set of real numbers κ_j. such that: