nemkommutatív

Nemkommutatív refers to a property of mathematical operations where the order of operands matters. An operation is called non-commutative if, for at least one pair of operands a and b, the result of applying the operation to a followed by b is different from applying the operation to b followed by a. This is often expressed as a * b ≠ b * a, where '*' represents the non-commutative operation.

Many fundamental mathematical operations are non-commutative. For example, matrix multiplication is a classic instance of a non-commutative operation. If you have two matrices A and B, the product AB will generally not be equal to the product BA. Similarly, quaternion multiplication is also non-commutative.

In contrast, commutative operations are those where the order of operands does not affect the result, meaning a * b = b * a. Addition and multiplication of ordinary numbers are prime examples of commutative operations.

The concept of non-commutativity is central to various fields of mathematics, including abstract algebra and linear algebra. It is a key characteristic that distinguishes different algebraic structures and plays a significant role in understanding their properties and behaviors. The study of non-commutative algebra has led to the development of powerful mathematical tools and theories with applications in physics, computer science, and other disciplines.