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. Given a complex number ${\displaystyle z=a+bi}$ (where a and b are real numbers), the complex conjugate of ${\displaystyle z}$, often denoted as ${\displaystyle {\overline {z}}}$, is equal to ${\displaystyle a-bi.}$[1][2][3]

Geometric representation (Argand diagram) of z and its conjugate z̅ in the complex plane. The complex conjugate is found by reflecting z across the real axis.

In polar form, the conjugate of ${\displaystyle re^{i\varphi }}$ is ${\displaystyle re^{-i\varphi }}$. This can be shown using Euler's formula.

The product of a complex number and its conjugate is a real number: ${\displaystyle a^{2}+b^{2}}$ (or ${\displaystyle r^{2}}$ in polar coordinates).

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

Notation

The complex conjugate of a complex number ${\displaystyle z}$ is written as ${\displaystyle {\overline {z}}}$ or ${\displaystyle z^{*}\!}$.[1][2] 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 ${\displaystyle e^{i\varphi }+{\text{c.c.}}}$ means ${\displaystyle e^{i\varphi }+e^{-i\varphi }}$.

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.

For any two complex numbers w,z, conjugation is distributive over addition, subtraction, multiplication and division.[2]

${\displaystyle {\begin{aligned}{\overline {z+w}}&={\overline {z}}+{\overline {w}}\\{\overline {z-w}}&={\overline {z}}-{\overline {w}}\\{\overline {zw}}&={\overline {z}}\;{\overline {w}}\\{\overline {\left({\frac {z}{w}}\right)}}&={\frac {\overline {z}}{\overline {w}}},\quad {\text{if }}w\neq 0\\\end{aligned}}}$

Real numbers are the only fixed points of conjugation. A complex number is equal to its complex conjugate if its imaginary part is zero.

${\displaystyle {\begin{aligned}{\overline {z}}&=z~\Leftrightarrow ~z\in \mathbb {R} \\\end{aligned}}}$

Composition of conjugation with the modulus is equivalent to the modulus alone.

${\displaystyle {\begin{aligned}\left|{\overline {z}}\right|&=\left|z\right|\\\end{aligned}}}$

Conjugation is an involution; the conjugate of the conjugate of a complex number z is z.[2]

${\displaystyle {\begin{aligned}{\overline {\overline {z}}}&=z\\\end{aligned}}}$

The product of a complex number with its conjugate is equal to the square of the number's modulus. This allows easy computation of the multiplicative inverse of a complex number given in rectangular coordinates.

${\displaystyle {\begin{aligned}z{\overline {z}}&={\left|z\right|}^{2}\\z^{-1}&={\frac {\overline {z}}{{\left|z\right|}^{2}}},\quad \forall z\neq 0\end{aligned}}}$

Conjugation is commutative under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments.

${\displaystyle {\begin{aligned}{\overline {z^{n}}}&=\left({\overline {z}}\right)^{n},\quad \forall n\in \mathbb {Z} \\\end{aligned}}}$

${\displaystyle \exp \left({\overline {z}}\right)={\overline {\exp(z)}}\,\!}$

${\displaystyle \log \left({\overline {z}}\right)={\overline {\log(z)}}\,\!}$ if z is non-zero

If ${\displaystyle p}$ is a polynomial with real coefficients, and ${\displaystyle p(z)=0}$, then ${\displaystyle p\left({\overline {z}}\right)=0}$ as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (see Complex conjugate root theorem).

In general, if ${\displaystyle \varphi \,}$ is a holomorphic function whose restriction to the real numbers is real-valued, and ${\displaystyle \varphi (z)}$ and ${\displaystyle \varphi ({\overline {z}})}$ are defined, then

${\displaystyle \varphi \left({\overline {z}}\right)={\overline {\varphi (z)}}.\,\!}$

The map ${\displaystyle \sigma (z)={\overline {z}}\,}$ from ${\displaystyle \mathbb {C} \,}$ to ${\displaystyle \mathbb {C} }$ is a homeomorphism (where the topology on ${\displaystyle \mathbb {C} }$ is taken to be the standard topology) and antilinear, if one considers ${\displaystyle \mathbb {C} \,}$ 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 ${\displaystyle \mathbb {C} /\mathbb {R} }$. This Galois group has only two elements: ${\displaystyle \sigma \,}$ and the identity on ${\displaystyle \mathbb {C} }$. Thus the only two field automorphisms of ${\displaystyle \mathbb {C} }$ that leave the real numbers fixed are the identity map and complex conjugation.

Use as a variable

Once a complex number ${\displaystyle z=x+yi}$ or ${\displaystyle z=re^{i\theta }}$ is given, its conjugate is sufficient to reproduce the parts of the z-variable:

• Real part: ${\displaystyle x=\operatorname {Re} (z)={\dfrac {z+{\overline {z}}}{2}}}$
• Imaginary part: ${\displaystyle y=\operatorname {Im} (z)={\dfrac {z-{\overline {z}}}{2i}}}$
• Modulus (or absolute value): ${\displaystyle r=\left|z\right|={\sqrt {z{\overline {z}}}}}$
• Argument: ${\displaystyle e^{i\theta }=e^{i\arg z}={\sqrt {\dfrac {z}{\overline {z}}}}}$, so ${\displaystyle \theta =\arg z={\dfrac {1}{i}}\ln {\sqrt {\frac {z}{\overline {z}}}}={\dfrac {\ln z-\ln {\overline {z}}}{2i}}}$

Furthermore, ${\displaystyle {\overline {z}}}$ can be used to specify lines in the plane: the set

${\displaystyle \left\{z\mid z{\overline {r}}+{\overline {z}}r=0\right\}}$

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

${\displaystyle {\frac {z-z_{0}}{{\overline {z}}-{\overline {z_{0}}}}}=u^{2}}$

determines the line through ${\displaystyle z_{0}}$ parallel to the line through 0 and 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, ${\textstyle {\overline {\mathbf {AB} }}=\left({\overline {\mathbf {A} }}\right)\left({\overline {\mathbf {B} }}\right)}$, where ${\textstyle {\overline {\mathbf {A} }}}$ represents the element-by-element conjugation of ${\displaystyle \mathbf {A} }$.[4] Contrast this to the property ${\textstyle \left(\mathbf {AB} \right)^{*}=\mathbf {B} ^{*}\mathbf {A} ^{*}}$, where ${\textstyle \mathbf {A} ^{*}}$ represents the conjugate transpose of ${\textstyle \mathbf {A} }$.

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 ${\textstyle a+bi+cj+dk}$ is ${\textstyle a-bi-cj-dk}$.

All these generalizations are multiplicative only if the factors are reversed:

${\displaystyle {\left(zw\right)}^{*}=w^{*}z^{*}.}$

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 ${\textstyle V}$ over the complex numbers. In this context, any antilinear map ${\textstyle \varphi :V\rightarrow V\,}$ that satisfies

1. ${\displaystyle \varphi ^{2}=\operatorname {id} _{V}\,}$, where ${\displaystyle \varphi ^{2}=\varphi \circ \varphi }$ and ${\displaystyle \operatorname {id} _{V}}$ is the identity map on ${\displaystyle V\,}$,
2. ${\displaystyle \varphi (zv)={\overline {z}}\varphi (v)}$ for all ${\displaystyle v\in V\,}$, ${\displaystyle z\in \mathbb {C} \,}$, and
3. ${\displaystyle \varphi \left(v_{1}+v_{2}\right)=\varphi \left(v_{1}\right)+\varphi \left(v_{2}\right)\,}$ for all ${\displaystyle v_{1}\in V\,}$, ${\displaystyle v_{2}\in V\,}$,

is called a complex conjugation, or a real structure. As the involution ${\displaystyle \varphi }$ is antilinear, it cannot be the identity map on ${\displaystyle V}$.

Of course, ${\textstyle \varphi }$ is a ${\textstyle \mathbb {R} }$-linear transformation of ${\textstyle V}$, 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 ${\displaystyle V}$.[5]

One example of this notion is the conjugate transpose operation of complex matrices defined above. Note that on generic complex vector spaces, there is no canonical notion of complex conjugation.

See also

• Absolute square
• Complex conjugate line
• Complex conjugate representation
• Complex conjugate vector space
• Composition algebra
• Conjugate (square roots)
• Wirtinger derivatives

References

1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-31.
2. Weisstein, Eric W. "Complex Conjugate". mathworld.wolfram.com. Retrieved 2020-08-31.
3. "Complex Numbers". www.mathsisfun.com. Retrieved 2020-08-31.
4. Arfken, Mathematical Methods for Physicists, 1985, pg. 201
5. Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29

Bibliography

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