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Complex conjugate

File:Complex conjugate picture.svg
Geometric representation of z and its conjugate in the complex plane. The complex conjugate is found by reflecting z across the real axis.

In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign.[1][2] For example, the complex conjugate of 3 + 4i is 3 − 4i.

In polar form, the conjugate of <math>\rho e^{i \phi}</math> is <math>\rho e^{-i \phi}</math>. This can be shown using Euler's formula.

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 conjugate of the some complex number <math>z</math> is written as <math>\overline z</math> or <math>z^*\!</math>. The first notation 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 <math>e^{i \phi}+c.c.</math> means <math>e^{i \phi}+e^{-i \phi}</math>

Properties

A significant property of the complex conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero or <math>-i</math>. The second case holds because the imaginary unit <math>i</math> is qualitatively indistinct from its additive and multiplicative inverse <math>-i</math>, as they both satisfy the definition for the imaginary unit: <math> x^2=-1</math>.

The following properties apply for all complex numbers z and w, unless stated otherwise, and can be proven by writing z and w in the form a + ib.

<math>\overline{(z + w)} = \overline{z} + \overline{w} \!\ </math>
<math>\overline{z-w} = \overline{z} - \overline{w} \!\ </math>
<math>\overline{(zw)} = \overline{z}\; \overline{w} \!\ </math>
<math>\overline{(z/w)} = \overline{z}/\overline{w} \!\ </math> if w is nonzero
<math>\overline{z} = z \!\ </math> if and only if z is real
<math>\overline{z^n} = (\overline{z})^n </math> for any integer n
<math>\left| \overline{z} \right| = \left| z \right|</math>
<math>{\left| z \right|}^2 = z\overline{z} = \overline{z}z</math>
<math>\overline{\overline{z}} = z \!\ </math>, involution (i.e., the conjugate of the conjugate of a complex number z is z)
<math>z^{-1} = \frac{\overline{z}}Template:\left{2}</math>
  • Imaginary part: <math>y = \operatorname{Im}\,(z) = \dfrac{z - \overline{z}}{2i}</math>
  • Modulus/absolute value: <math>\rho = \left| z \right| = \sqrt {z \overline{z}}</math>
  • Argument: <math>e^{i\theta} = e^{i\arg z} = \sqrt {\dfrac{z}{\overline z}}</math>, so <math>\theta = \arg z = \dfrac{1}{i}\ln \sqrt{\frac{z}{\overline z}} = \dfrac{\ln z - \ln \overline z}{2i}</math>

Thus the pair of variables <math>z\,</math> and <math>\overline{z}</math> also serve up the plane as do x,y and <math>\rho \,</math> and <math>\theta</math>. Furthermore, the <math>\overline{z}</math> variable is useful in specifying lines in the plane:

<math> \{z \mid z \overline{r} + \overline{z} r = 0 \}</math>

is a line through the origin and perpendicular to <math>\overline{r}</math> since the real part of <math>z\cdot\overline{r}</math> is zero only when the cosine of the angle between <math>z\,</math> and <math>\overline{r}</math> is zero. Similarly, for a fixed complex unit u = exp(b i), the equation:

<math> \frac{z - z_0}{\overline{z} - \overline{z_0}} = u</math>

determines the line through <math>z_0\,</math> 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 explicated by use of complex conjugation.

For matrices of complex numbers <math>\overline{\mathbf{AB}} = (\overline{\mathbf{A}}) (\overline{\mathbf{B}})</math>, where <math>\overline{\mathbf{A}}</math> represents the element-by-element conjugation of <math>\mathbf{A}</math>.[3] Contrast this to the property <math>(\mathbf{AB})^*=\mathbf{B}^* \mathbf{A}^* </math>, where <math>\mathbf{A}^*</math> represents the conjugate transpose of <math>\mathbf{A}</math>.

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 coquaternions: the conjugate of <math>a + bi + cj + dk</math> is <math>a - bi - cj - dk</math>.

Note that all these generalizations are multiplicative only if the factors are reversed:

<math>{\left(zw\right)}^* = w^* z^*.</math>

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 <math>V</math> over the complex numbers. In this context, any antilinear map <math>\phi: V \rightarrow V\,</math> that satisfies

  1. <math>\phi^2 = \operatorname{id}_V\,</math>, where <math>\phi^2=\phi\circ\phi</math> and <math>\operatorname{id}_V\,</math> is the identity map on <math>V\,</math>,
  2. <math>\phi(zv) = \overline{z} \phi(v)</math> for all <math>v\in V\,</math>, <math>z\in{\mathbb C}\,</math>, and
  3. <math>\phi(v_1+v_2) = \phi(v_1)+\phi(v_2)\,</math> for all <math>v_1\in V\,</math>, <math>v_2\in V\,</math>,

is called a complex conjugation, or a real structure. As the involution <math>\operatorname{\phi}</math> is antilinear, it cannot be the identity map on <math>V</math>. Of course, <math>\operatorname{\phi}</math> is a <math>\mathbb{R}</math>-linear transformation of <math>V</math>, 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 <math>V</math>.[4] 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.

See also

Notes

  1. ^ Weisstein, Eric W., "Complex Conjugates", MathWorld.
  2. ^ Weisstein, Eric W., "Imaginary Numbers", MathWorld.
  3. ^ Arfken, Mathematical Methods for Physicists, 1985, pg. 201
  4. ^ Budinich, P. and Trautman, A. The Spinorial Chessboard. Spinger-Verlag, 1988, p. 29

References

  • Budinich, P. and Trautman, A. The Spinorial Chessboard. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).
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