Conjugately

Conjugately is a term used in mathematics, particularly in algebra and number theory, to describe a relationship between two entities that are formed by a specific transformation. The most common usage of the term "conjugate" refers to complex numbers. For a complex number of the form a + bi, where 'a' is the real part and 'b' is the imaginary part, its conjugate is a - bi. This involves changing the sign of the imaginary component.

In linear algebra, the concept of a conjugate transpose is important for matrices, especially those with complex entries. The conjugate transpose of a matrix, often denoted by A*, is obtained by taking the transpose of the matrix and then taking the complex conjugate of each entry. This operation is fundamental in defining Hermitian matrices, which are equal to their own conjugate transpose.

The idea of conjugation also extends to elements within algebraic structures like groups and fields. In group theory, two elements 'g' and 'h' in a group G are said to be conjugate if there exists an element 'x' in G such that h = xgx⁻¹. This defines an equivalence relation where each element belongs to a conjugacy class. In field extensions, conjugate elements are roots of the same minimal polynomial over a base field. The term "conjugately" therefore signifies this symmetrical or related pairing based on specific mathematical operations.