Inequality (mathematics) - Vector Inequalities

Vector Inequalities

Inequality relationships similar to those defined above can also be defined for column vector. If we let the vectors (meaning that and where and are real numbers for ), we can define the following relationships.

• if for
• if for
• if for and
• if for

Similarly, we can define relationships for, and . We note that this notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).

We observe that the property of Trichotomy (as stated above) is not valid for vector relationships. We consider the case where and . There exists no valid inequality relationship between these two vectors. Also, a multiplicative inverse would need to be defined on a vector before this property could be considered. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.