Complex conjugate
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.[1] For example, (if a and b are real, then) the complex conjugate of is
In polar form, the conjugate of is . This can be shown using Euler's formula.
The product of a complex number and its conjugate is a real number: or in polar coordinates.
Complex conjugates are important for finding roots of polynomials. According to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients (such as the quadratic equation or the cubic equation), so is its conjugate.
Notation
The complex conjugate of a complex number is written as or . The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate transpose, while the bar-notation is more common in pure mathematics. If a complex number is represented as a 2×2 matrix, the notations are identical. In some texts, the complex conjugate of a previous known number is abbreviated as "c.c.". For example, writing means .
Properties
The following properties apply for all complex numbers z and w, unless stated otherwise, and can be proved by writing z and w in the form a + bi.
A significant property of the complex conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the complex number is real.
For any two complex numbers w,z:
The second to last relation is involution; i.e., the conjugate of the conjugate of a complex number z is z. The last is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
- if z is non-zero
If is a polynomial with real coefficients, and , then as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (see Complex conjugate root theorem).
In general, if is a holomorphic function whose restriction to the real numbers is real-valued, and is defined, then
The map from to is a homeomorphism (where the topology on is taken to be the standard topology) and antilinear, if one considers as a complex vector space over itself. Even though it appears to be a 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.
Use as a variable
Once a complex number or is given, its conjugate is sufficient to reproduce the parts of the z-variable:
- Real part:
- Imaginary part:
- Modulus (or absolute value):
- Argument: , so
Furthermore, can be used to specify lines in the plane: the set
is a line through the origin and perpendicular to since the real part of is zero only when the cosine of the angle between and is zero. Similarly, for a fixed complex unit u = exp(b i), the equation
determines the line through in the direction of u.
These uses of the conjugate of z as a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley.
Generalizations
The other planar real algebras, dual numbers, and split-complex numbers are also analyzed using complex conjugation.
For matrices of complex numbers , where represents the element-by-element conjugation of .[2] Contrast this to the property , where represents the conjugate transpose of .
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 *-operations of C*-algebras.
One may also define a conjugation for quaternions and split-quaternions: the conjugate of is .
All these generalizations are multiplicative only if the factors are reversed:
Since the multiplication of planar real algebras 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 antilinear map that satisfies
- , where and is the identity map on ,
- for all , , and
- for all , ,
is called a complex conjugation, or a real structure. As the involution is antilinear, it cannot be the identity map on .
Of course, is a -linear transformation of , if one notes that every complex space V has a real form obtained by taking the same vectors as in the original space and restricting the scalars to be real. The above properties actually define a real structure on the complex vector space .[3]
One example of this notion is the conjugate transpose operation of complex matrices defined above. It should be remarked that on generic complex vector spaces there is no canonical notion of complex conjugation.
Notes
- Weisstein, Eric W. "Imaginary Numbers". MathWorld.
- Arfken, Mathematical Methods for Physicists, 1985, pg. 201
- Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29
References
- Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).