Movatterモバイル変換

[0]ホーム

Jump to content

Complex conjugate

Edit links

From Wikipedia, the free encyclopedia

Fundamental operation on complex numbers

Geometric representation (Argand diagram) of $z {\displaystyle z}$ and its conjugate ${\overline {z}}$ in the complex plane. The complex conjugate is found byreflecting $z {\displaystyle z}$ across the real axis.

Geometric representation (Argand diagram) of $z {\displaystyle z}$ and its conjugate ${\overline {z}}$ in the complex plane. The complex conjugate is found byreflecting $z {\displaystyle z}$ across the real axis.

Inmathematics, thecomplex conjugate of acomplex number is the number with an equalreal part and animaginary part equal inmagnitude but opposite insign. That is, if $a {\displaystyle a}$ and $b {\displaystyle b}$ are real numbers, then the complex conjugate of $a+bi$ is $a-bi.$ The complex conjugate of $z {\displaystyle z}$ is often denoted as ${\overline {z}}$ or $z^{*}$ .

Inpolar form, if $r {\displaystyle r}$ and $\varphi$ are real numbers then the conjugate of $re^{i\varphi }$ is $re^{-i\varphi }.$ This can be shown usingEuler's formula.

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

If a root of aunivariate polynomial with real coefficients is complex, then itscomplex conjugate is also a root.

Notation

[edit]

The complex conjugate of a complex number $z {\displaystyle z}$ is written as ${\overline {z}}$ or $z^{*}.$ The first notation, avinculum, avoids confusion with the notation for theconjugate transpose of amatrix, which can be thought of as a generalization of the complex conjugate. The second is preferred inphysics, wheredagger (†) is used for the conjugate transpose, as well as electrical engineering andcomputer engineering, where bar notation can be confused for thelogical negation ("NOT")Boolean algebra symbol, while the bar notation is more common inpure mathematics.

If a complex number isrepresented as a $2\times 2$ matrix, the notations are identical, and the complex conjugate corresponds to thematrix transpose, which is a flip along the diagonal.^[1]

Properties

[edit]

The following properties apply for all complex numbers $z {\displaystyle z}$ and $w, {\displaystyle w,}$ unless stated otherwise, and can be proved by writing $z {\displaystyle z}$ and $w {\displaystyle w}$ in the form $a+bi.$

For any two complex numbers, conjugation isdistributive over addition, subtraction, multiplication and division:^[2] ${\begin{aligned}{\overline {z+w}}&={\overline {z}}+{\overline {w}},\\{\overline {z-w}}&={\overline {z}}-{\overline {w}},\\{\overline {zw}}&={\overline {z}}\;{\overline {w}},\quad {\text{and}}\\{\overline {\left({\frac {z}{w}}\right)}}&={\frac {\overline {z}}{\overline {w}}},\quad {\text{if }}w\neq 0.\end{aligned}}$

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

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

Conjugation is aninvolution, that is, the conjugate of the conjugate of a complex number $z {\displaystyle z}$ is $z . {\displaystyle z.}$ In symbols, ${\overline {\overline {z}}}=z.$ ^[2]

The product of a complex number with its conjugate is equal to the square of the number's modulus: $z{\overline {z}}={\left|z\right|}^{2}.$ This allows easy computation of themultiplicative inverse of a complex number given in rectangular coordinates: $z^{-1}={\frac {\overline {z}}{{\left|z\right|}^{2}}},\quad {\text{ for all }}z\neq 0.$

Conjugation iscommutative under composition with exponentiation to integer powers, with the exponential function, and with the natural logarithm for nonzero arguments: ${\overline {z^{n}}}=\left({\overline {z}}\right)^{n},\quad {\text{ for all }}n\in \mathbb {Z}$ ^{[note 1]} $\exp \left({\overline {z}}\right)={\overline {\exp(z)}}$ $\ln \left({\overline {z}}\right)={\overline {\ln(z)}}{\text{ if }}z{\text{ is not zero or a negative real number }}$

If $p {\displaystyle p}$ is apolynomial withreal 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 (seeComplex conjugate root theorem).

In general, if $\varphi$ is aholomorphic 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 ahomeomorphism (where the topology on $\mathbb {C}$ is taken to be the standard topology) andantilinear, if one considers $\mathbb {C}$ as a complexvector space over itself. Even though it appears to be awell-behaved function, it is notholomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It isbijective and compatible with the arithmetical operations, and hence is afield automorphism. As it keeps the real numbers fixed, it is an element of theGalois group of thefield 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

[edit]

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

These uses of the conjugate of $z {\displaystyle z}$ as a variable are illustrated inFrank Morley's bookInversive Geometry (1933), written with his son Frank Vigor Morley.

Generalizations

[edit]

The other planar real unital algebras,dual numbers, andsplit-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 $\mathbf {A} .$ ^[3] Contrast this to the property ${\textstyle \left(\mathbf {AB} \right)^{*}=\mathbf {B} ^{*}\mathbf {A} ^{*},}$ where ${\textstyle \mathbf {A} ^{*}}$ represents theconjugate transpose of ${\textstyle \mathbf {A} .}$

Taking theconjugate transpose (or adjoint) of complexmatrices generalizes complex conjugation. Even more general is the concept ofadjoint operator for operators on (possibly infinite-dimensional) complexHilbert spaces. All this is subsumed by the *-operations ofC*-algebras.

One may also define a conjugation forquaternions andsplit-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 iscommutative, this reversal is not needed there.

There is also an abstract notion of conjugation forvector spaces ${\textstyle V}$ over thecomplex numbers. In this context, anyantilinear map ${\textstyle \varphi :V\to V}$ that satisfies

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

is called acomplex conjugation, or areal structure. As the involution $\varphi$ isantilinear, it cannot be the identity map on $V . {\displaystyle V.}$

Of course, ${\textstyle \varphi }$ is a ${\textstyle \mathbb {R} }$ -linear transformation of ${\textstyle V,}$ if one notes that every complex space $V {\displaystyle V}$ has a real form obtained by taking the samevectors 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 $V . {\displaystyle V.}$ ^[4]

One example of this notion is the conjugate transpose operation of complex matrices defined above. However, on generic complex vector spaces, there is nocanonical notion of complex conjugation.

References

[edit]

^"Lesson Explainer: Matrix Representation of Complex Numbers | Nagwa".www.nagwa.com. Retrieved2023-01-04.
^^a ^bFriedberg, Stephen; Insel, Arnold; Spence, Lawrence (2018),Linear Algebra (5 ed.), Pearson,ISBN 978-0134860244, Appendix D
^Arfken,Mathematical Methods for Physicists, 1985, pg. 201
^Budinich, P. and Trautman, A.The Spinorial Chessboard. Springer-Verlag, 1988, p. 29