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Complex conjugate
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{{Short description|Fundamental operation on complex numbers}} [[File:Complex conjugate picture.svg|thumb|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 [[Reflection symmetry|reflecting]] <math>z</math> across the real axis.]] In [[mathematics]], the '''complex conjugate''' of a [[complex number]] is the number with an equal [[Real number|real]] part and an [[Imaginary number|imaginary]] part equal in [[Magnitude of a complex number|magnitude]] but opposite in [[Sign (mathematics)|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 coordinate system#Complex numbers|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 coordinate system|polar coordinates]]). If a root of a [[univariate]] polynomial with real coefficients is complex, then its [[Complex conjugate root theorem|complex conjugate is also a root]]. == Notation == 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 (symbol)|vinculum]], avoids confusion with the notation for the [[conjugate transpose]] of a [[Matrix (mathematics)|matrix]], which can be thought of as a generalization of the complex conjugate. The second is preferred in [[physics]], where [[Dagger (mark)|dagger]] (β ) 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 [[Complex number#Matrix representation of complex numbers|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>{{Cite web |title=Lesson Explainer: Matrix Representation of Complex Numbers {{!}} Nagwa |url=https://www.nagwa.com/en/explainers/152196980513/ |access-date=2023-01-04 |website=www.nagwa.com |language=en}}</ref> == Properties == 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 property|distributive]] over addition, subtraction, multiplication and division:<ref name = fis>{{citation|title = Linear Algebra | first1 = Stephen | last1 = Friedberg | first2 = Arnold | last2 = Insel | first3 = Lawrence | last3 =Spence | edition = 5 | year = 2018 | publisher = Pearson | isbn = 978-0134860244}}, 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 point (mathematics)|fixed point]]s of conjugation. Conjugation does not change the modulus of a complex number: <math>\left| \overline{z} \right| = |z|.</math> Conjugation is an [[involution (mathematics)|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}}{{\left| z \right|}^2},\quad \text{ for all } z \neq 0.</math> Conjugation is [[commutative]] under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments: <math display="block">\overline{z^n} = \left(\overline{z}\right)^n,\quad \text{ for all } n \in \Z </math><ref group =note>See [[Exponentiation#Non-integer powers of complex numbers]].</ref> <math display="block">\exp\left(\overline{z}\right) = \overline{\exp(z)}</math> <math display="block">\ln\left(\overline{z}\right) = \overline{\ln(z)} \text{ if } z \text{ is not zero or a negative real number }</math> If <math>p</math> is a [[polynomial]] with [[real number|real]] coefficients and <math>p(z) = 0,</math> then <math>p\left(\overline{z}\right) = 0</math> as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs (''see'' [[Complex conjugate root theorem]]). In general, if <math>\varphi</math> is a [[holomorphic function]] whose restriction to the real numbers is real-valued, and <math>\varphi(z)</math> and <math>\varphi(\overline{z})</math> are defined, then <math display="block">\varphi\left(\overline{z}\right) = \overline{\varphi(z)}.\,\!</math> The map <math>\sigma(z) = \overline{z}</math> from <math>\Complex</math> to <math>\Complex</math> is a [[homeomorphism]] (where the topology on <math>\Complex</math> is taken to be the standard topology) and [[antilinear]], if one considers <math>\Complex</math> as a complex [[vector space]] over itself. Even though it appears to be a [[well-behaved]] function, it is not [[holomorphic function|holomorphic]]; it reverses orientation whereas holomorphic functions locally preserve orientation. It is [[bijective]] and compatible with the arithmetical operations, and hence is a [[field (mathematics)|field]] [[automorphism]]. As it keeps the real numbers fixed, it is an element of the [[Galois group]] of the [[field extension]] <math>\Complex/\R.</math> This Galois group has only two elements: <math>\sigma</math> and the identity on <math>\Complex.</math> Thus the only two field automorphisms of <math>\Complex</math> that leave the real numbers fixed are the identity map and complex conjugation. ==Use as a variable== Once a complex number <math>z = x + yi</math> or <math>z = re^{i\theta}</math> is given, its conjugate is sufficient to reproduce the parts of the <math>z</math>-variable: * Real part: <math>x = \operatorname{Re}(z) = \dfrac{z + \overline{z}}{2}</math> * Imaginary part: <math>y = \operatorname{Im}(z) = \dfrac{z - \overline{z}}{2i}</math> * [[Absolute value|Modulus (or absolute value)]]: <math>r= \left| z \right| = \sqrt{z\overline{z}}</math> * [[Argument (complex analysis)|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. ==Generalizations== The other planar real unital algebras, [[dual numbers]], and [[split-complex number]]s 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 [[Matrix (mathematics)|matrices]] generalizes complex conjugation. Even more general is the concept of [[adjoint operator]] for operators on (possibly infinite-dimensional) complex [[Hilbert space]]s. All this is subsumed by the *-operations of [[C*-algebra]]s. One may also define a conjugation for [[quaternion]]s and [[split-quaternion]]s: 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 number]]s. 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 {{em|complex conjugation}}, 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 [[vector (mathematics and physics)|vector]]s 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 {{em|[[Canonical form|canonical]]}} notion of complex conjugation. ==See also== * {{annotated link|Absolute square}} * {{annotated link|Complex conjugate line}} * {{annotated link|Complex conjugate representation}} * {{annotated link|Complex conjugate vector space}} * {{annotated link|Composition algebra}} * {{annotated link|Conjugate (square roots)}} * {{annotated link|Hermitian function}} * {{annotated link|Wirtinger derivatives}} ==References== {{reflist}} == Footnotes == {{reflist|group=note}} ==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). {{Complex numbers}} {{DEFAULTSORT:Complex Conjugate}} [[Category:Complex numbers]]
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