# Complex conjugate

Last updated Geometric representation (Argand diagram) of z{\displaystyle z} and its conjugate z¯{\displaystyle {\overline {z}}} in the complex plane. The complex conjugate is found by reflecting z{\displaystyle z} 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 $a$ and $b$ are real, then) the complex conjugate of $a+bi$ is equal to $a-bi.$ The complex conjugate of $z$ is often denoted as ${\overline {z}}.$ ## Contents

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

The product of a complex number and its conjugate is a real number: $a^{2}+b^{2}$ (or $r^{2}$ 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 $z$ is written as ${\overline {z}}$ or $z^{*}.$ 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 $2\times 2$ matrix, the notations are identical.[ clarification needed ]

## 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, conjugation is distributive over addition, subtraction, multiplication and division: 

{\begin{aligned}{\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{aligned}} A complex number is equal to its complex conjugate if its imaginary part is zero, or equivalently, 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: $\left|{\overline {z}}\right|=|z|.$ Conjugation is an involution, that is, the conjugate of the conjugate of a complex number $z$ is $z.$ In symbols, ${\overline {\overline {z}}}=z.$ 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.

{\begin{aligned}z{\overline {z}}&={\left|z\right|}^{2}\\z^{-1}&={\frac {\overline {z}}{{\left|z\right|}^{2}}},\quad {\text{ for all }}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:

${\overline {z^{n}}}=\left({\overline {z}}\right)^{n},\quad {\text{ for all }}n\in \mathbb {Z}$ $\exp \left({\overline {z}}\right)={\overline {\exp(z)}}$ $\ln \left({\overline {z}}\right)={\overline {\ln(z)}}{\text{ if }}z{\text{ is non-zero }}$ If $p$ is a polynomial with real coefficients and $p(z)=0,$ then $p\left({\overline {z}}\right)=0$ as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (see Complex conjugate root theorem).

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

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

## Use as a variable

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

• Real part: $x=\operatorname {Re} (z)={\dfrac {z+{\overline {z}}}{2}}$ • Imaginary part: $y=\operatorname {Im} (z)={\dfrac {z-{\overline {z}}}{2i}}$ • Modulus (or absolute value): $r=\left|z\right|={\sqrt {z{\overline {z}}}}$ • Argument: $e^{i\theta }=e^{i\arg z}={\sqrt {\dfrac {z}{\overline {z}}}},$ so $\theta =\arg z={\dfrac {1}{i}}\ln {\sqrt {\frac {z}{\overline {z}}}}={\dfrac {\ln z-\ln {\overline {z}}}{2i}}$ Furthermore, ${\overline {z}}$ can be used to specify lines in the plane: the set

$\left\{z:z{\overline {r}}+{\overline {z}}r=0\right\}$ is a line through the origin and perpendicular to ${r},$ since the real part of $z\cdot {\overline {r}}$ is zero only when the cosine of the angle between $z$ and ${r}$ is zero. Similarly, for a fixed complex unit $u=e^{ib},$ the equation

${\frac {z-z_{0}}{{\overline {z}}-{\overline {z_{0}}}}}=u^{2}$ determines the line through $z_{0}$ 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 $\mathbf {A} .$ 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:

${\left(zw\right)}^{*}=w^{*}z^{*}.$ 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 ${\textstyle \varphi :V\to V}$ that satisfies

1. $\varphi ^{2}=\operatorname {id} _{V}\,,$ where $\varphi ^{2}=\varphi \circ \varphi$ and $\operatorname {id} _{V}$ is the identity map on $V,$ 2. $\varphi (zv)={\overline {z}}\varphi (v)$ for all $v\in V,z\in \mathbb {C} ,$ and
3. $\varphi \left(v_{1}+v_{2}\right)=\varphi \left(v_{1}\right)+\varphi \left(v_{2}\right)\,$ for all $v_{1}v_{2},\in V,$ is called a complex conjugation, or a real structure. As the involution $\varphi$ is antilinear, it cannot be the identity map on $V.$ 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 $V.$ One example of this notion is the conjugate transpose operation of complex matrices defined above. However, on generic complex vector spaces, there is no canonical notion of complex conjugation.

## Related Research Articles

In quantum mechanics, bra–ket notation, or Dirac notation, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets". In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. In mathematics, an inner product space or a Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors, often denoted using angle brackets. Inner products allow the rigorous introduction of intuitive geometrical notions, such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors. Inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.

Riesz representation theorem, sometimes called Riesz–Fréchet representation theorem, named after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural one as will be described next; a natural isomorphism.

The Cauchy–Schwarz inequality is considered one of the most important and widely used inequalities in mathematics. In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.

In mathematics, particularly in linear algebra, a skew-symmetricmatrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In mathematics, a linear form is a linear map from a vector space to its field of scalars. In particle physics, the Georgi–Glashow model is a particular grand unified theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974. In this model the standard model gauge groups SU(3) × SU(2) × U(1) are combined into a single simple gauge group SU(5). The unified group SU(5) is then thought to be spontaneously broken into the standard model subgroup below a very high energy scale called the grand unification scale.

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

In mathematics, the total variation identifies several slightly different concepts, related to the structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation xf(x), for x ∈ [a, b]. Functions whose total variation is finite are called functions of bounded variation.

In mathematics, the complexification of a vector space V over the field of real numbers yields a vector space VC over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for V may also serve as a basis for VC over the complex numbers.

In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector.

A radial basis function (RBF) is a real-valued function whose value depends only on the distance between the input and some fixed point, either the origin, so that , or some other fixed point , called a center, so that . Any function that satisfies the property is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection which forms a basis for some function space of interest, hence the name. In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

In applied mathematics, polyharmonic splines are used for function approximation and data interpolation. They are very useful for interpolating and fitting scattered data in many dimensions. Special cases include thin plate splines and natural cubic splines in one dimension. In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.

In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map , with , giving the "canonical" real structure on , that is .

In probability theory, the family of complex normal distributions characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance matrix , and the relation matrix . The standard complex normal is the univariate distribution with , , and .

1. Friedberg, Stephen; Insel, Arnold; Spence, Lawrence (2018), Linear Algebra (5 ed.), ISBN   978-0134860244 , Appendix D
2. Arfken, Mathematical Methods for Physicists, 1985, pg. 201
3. Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988, p. 29