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In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
In mathematics, the conjugate of an expression of the form + is , provided that does not appear in a and b.One says also that the two expressions are conjugate. In particular, the two solutions of a quadratic equation are conjugate, as per the in the quadratic formula =.
Conjugate transpose, the complex conjugate of the transpose of a matrix; Harmonic conjugate in complex analysis; Conjugate (graph theory), an alternative term for a line graph, i.e. a graph representing the edge adjacencies of another graph; In group theory, various notions are called conjugation: Inner automorphism, a type of conjugation ...
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of + being , for real numbers and ).
Geometric representation of z and its conjugate z in the complex plane. The complex conjugate of the complex number z = x + yi is defined as ¯ =. [11] It is also denoted by some authors by . Geometrically, z is the "reflection" of z about the real axis.
In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial p K,α (x) of α over K. Conjugate elements are commonly called conjugates in contexts where this is not ambiguous.
In mathematics, the complex conjugate of a complex vector space is a complex vector space ¯ that has the same elements and additive group structure as , but whose scalar multiplication involves conjugation of the scalars.
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P. [1]