Complex conjugate

Geometric representation (Argand diagram) of <math>z</math> and its conjugate <math>\overline{z}</math> in the complex plane. The complex conjugate is found by reflecting <math>z</math> across the real axis.

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 <math>a</math> and <math>b</math> are real numbers, then the complex conjugate of <math> a + bi</math> is <math>a - bi.</math> The complex conjugate of <math>z</math> is often denoted as <math>\overline{z}</math> or <math>z^*</math>.

In polar form, if <math>r</math> and <math>\varphi</math> are real numbers then the conjugate of <math>r e^{i \varphi}</math> is <math>r e^{-i \varphi}.</math> This can be shown using Euler's formula.

The product of a complex number and its conjugate is a real number: <math>a^2 + b^2</math> (or <math>r^2</math> in polar coordinates).

If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.

NotationEdit

The complex conjugate of a complex number <math>z</math> is written as <math>\overline z</math> or <math>z^*.</math> 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, as well as electrical engineering and computer engineering, where bar notation can be confused for the logical negation ("NOT") Boolean algebra symbol, while the bar notation is more common in pure mathematics.

If a complex number is represented as a <math>2 \times 2</math> matrix, the notations are identical, and the complex conjugate corresponds to the matrix transpose, which is a flip along the diagonal.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

PropertiesEdit

The following properties apply for all complex numbers <math>z</math> and <math>w,</math> unless stated otherwise, and can be proved by writing <math>z</math> and <math>w</math> in the form <math>a + b i.</math>

For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division:<ref name = fis>Template:Citation, Appendix D</ref> <math display="block">\begin{align}

                    \overline{z + w} &= \overline{z} + \overline{w}, \\
                    \overline{z - w} &= \overline{z} - \overline{w}, \\
                       \overline{zw} &= \overline{z} \; \overline{w}, \quad \text{and} \\
 \overline{\left(\frac{z}{w}\right)} &= \frac{\overline{z}}{\overline{w}},\quad \text{if } w \neq 0.

\end{align}</math>

A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real. In other words, real numbers are the only fixed points of conjugation.

Conjugation does not change the modulus of a complex number: <math>\left| \overline{z} \right| = |z|.</math>

Conjugation is an involution, that is, the conjugate of the conjugate of a complex number <math>z</math> is <math>z.</math> In symbols, <math>\overline{\overline{z}} = z.</math><ref name = fis />

The product of a complex number with its conjugate is equal to the square of the number's modulus: <math display="block">z\overline{z} = {\left| z \right|}^2.</math> This allows easy computation of the multiplicative inverse of a complex number given in rectangular coordinates: <math display="block">z^{-1} = \frac{\overline{z}}Template:\left{2}</math>

Imaginary part: <math>y = \operatorname{Im}(z) = \dfrac{z - \overline{z}}{2i}</math>
Modulus (or absolute value): <math>r= \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>

Furthermore, <math>\overline{z}</math> can be used to specify lines in the plane: the set <math display="block">\left\{z : z \overline{r} + \overline{z} r = 0 \right\}</math> is a line through the origin and perpendicular to <math>{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>{r}</math> is zero. Similarly, for a fixed complex unit <math>u = e^{i b},</math> the equation <math display="block">\frac{z - z_0}{\overline{z} - \overline{z_0}} = u^2</math> determines the line through <math>z_0</math> parallel to the line through 0 and <math>u.</math>

These uses of the conjugate of <math>z</math> as a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley.

GeneralizationsEdit

The other planar real unital algebras, dual numbers, and split-complex numbers are also analyzed using complex conjugation.

For matrices of complex numbers, <math display="inline">\overline{\mathbf{AB}} = \left(\overline{\mathbf{A}}\right) \left(\overline{\mathbf{B}}\right),</math> where <math display="inline">\overline{\mathbf{A}}</math> represents the element-by-element conjugation of <math>\mathbf{A}.</math><ref>Arfken, Mathematical Methods for Physicists, 1985, pg. 201</ref> Contrast this to the property <math display="inline">\left(\mathbf{AB}\right)^*=\mathbf{B}^* \mathbf{A}^*,</math> where <math display="inline">\mathbf{A}^*</math> represents the conjugate transpose of <math display="inline">\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 split-quaternions: the conjugate of <math display="inline">a + bi + cj + dk</math> is <math display="inline">a - bi - cj - dk.</math>

All these generalizations are multiplicative only if the factors are reversed: <math display="block">{\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 display="inline">V</math> over the complex numbers. In this context, any antilinear map <math display="inline">\varphi: V \to V</math> that satisfies

<math>\varphi^2 = \operatorname{id}_V\,,</math> where <math>\varphi^2 = \varphi \circ \varphi</math> and <math>\operatorname{id}_V</math> is the identity map on <math>V,</math>
<math>\varphi(zv) = \overline{z} \varphi(v)</math> for all <math>v \in V, z \in \Complex,</math> and
<math>\varphi\left(v_1 + v_2\right) = \varphi\left(v_1\right) + \varphi\left(v_2\right)\,</math> for all <math>v_1, v_2 \in V,</math>

is called a Template:Em, or a real structure. As the involution <math>\varphi</math> is antilinear, it cannot be the identity map on <math>V.</math>

Of course, <math display="inline">\varphi</math> is a <math display="inline">\R</math>-linear transformation of <math display="inline">V,</math> if one notes that every complex space <math>V</math> 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><ref>Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29</ref>

One example of this notion is the conjugate transpose operation of complex matrices defined above. However, on generic complex vector spaces, there is no Template:Em notion of complex conjugation.

ReferencesEdit

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FootnotesEdit

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BibliographyEdit

Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. Template:ISBN. (antilinear maps are discussed in section 3.3).

Template:Complex numbers