|
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number (where and are real numbers) is The complex conjugate is also very commonly denoted by . Here we choose to avoid confusion with the notation for the conjugate transpose of a matrix (which can be thought of as a generalization of complex conjugation). Notice that if a complex number is treated as a vector, the notations are identical. For example, , and . One usually thinks of complex numbers as points in a plane with a cartesian coordinate system. The -axis contains the real numbers and the -axis contains the multiples of . In this view, complex conjugation corresponds to reflection at the x-axis. In polar form, however, the conjugate of is given by . This can easily be verified by using Euler's formula.
Properties These properties apply for all complex numbers and , unless stated otherwise. if is non-zero if and only if is real ight| = left| z ight| if is non-zero The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates. If is a polynomial with real coefficients, and , then as well. Thus non-real roots of real polynomials occur in complex conjugate pairs. The function from to is continuous. Even though it appears to be a "tame" well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the arithmetical operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension . This Galois group has only two elements: and the identity on . Thus the only two field automorphisms of that leave the real numbers fixed are the identity map and complex conjugation. Generalizations Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces. All this is subsumed by the One may also define a conjugation for quaternions: the conjugate of is . Note that all these generalizations are multiplicative only if the factors are reversed: Since the multiplication of complex numbers is commutative, this reversal is not needed there. There is also an abstract notion of conjugation for vector spaces over the complex numbers. In this context, any (real) linear transformation that satisfies is called a complex conjugation. One example of this notion is the conjugate transpose operation of complex matrices defined above. It should be remarked that on general complex vector spaces there is no canonical notion of complex conjugation. See also | ||||||||
|
| |||||||||
 |
|
| |