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Complex conjugate

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In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number <math>z=a+ib<math> (where a and b are real numbers) is defined to be <math>z^* = a - ib<math>. It is also often denoted by a bar over the number, rather than a star, which often is used also for the conjugate transpose. If a complex number is treated as a 1×1 vector, the notations are identical.

For example, <math>(3-2i)^* = 3 + 2i<math>, <math>i^* = -i<math> and <math>7^*=7<math>.

One usually thinks of complex numbers as points in a plane with a cartesian coordinate system. The x-axis contains the real numbers and the y-axis contains the multiples of i. In this view, complex conjugation corresponds to reflection at the x-axis.

Properties

These properties apply for all complex numbers z and w, unless stated otherwise.

<math>(z + w)^* = z^* + w^*<math>
<math>(zw)^* = z^* w^*<math>
<math>\left({\frac{z}{w}}\right)^* = \frac{z^*}{w^*}<math> if w is non-zero
<math>z^* = z<math> if and only if z is real
<math>\left| z^* \right| = \left| z \right|<math>
<math>{\left| z \right|}^2 = zz^*<math>
<math>z^{-1} = \frac{z^*}{{\left| z \right|}^2}<math> if z is non-zero

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

If p is a polynomial with real coefficients, and <math>p(z) = 0<math>, then <math>p(z^*) = 0<math> as well. Thus the roots of real polynomials outside of the real line occur in complex conjugate pairs.

The function <math>\phi(z) = z^*<math> from C to C is continuous. Even though it appears to be a "tame" 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 C / R. This Galois group has only two elements: φ and the identity on C. Thus the only two field automorphisms of C that leave the real numbers fixed are the identity map and complex conjugation.

Generalizations

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-star algebras.

One may also define a conjugation for quaternions: the conjugate of <math>a + bi + cj + dk<math> is <math>a - bi - cj - dk<math>.

Note that all these generalizations are multiplicative only if the factors are reversed:

<math>{\left(zw\right)}^* = w^* z^*.<math>

Since the multiplication of complex numbers is commutative, this reversal is "invisible" there.

There is also an abstract notion of conjugation for vector spaces <math>V<math> over the complex numbers. In this context, any (real) linear transformation <math>\phi: V \rightarrow V <math> that satisfies

  1. <math>\phi\neq id_V<math>, the identity function on <math>V<math>,
  2. <math>\phi^2 = id_V<math>, and
  3. <math>\phi(zv) = z^* \phi(v)<math> for all <math>v\in V<math>, <math>z\in{\mathbb C}<math>,

is called a complex conjugation. One example of this notion is the conjugate transpose operation of complex matrices defined above. It should be remarked that on general complex vector spaces there is no canonical notion of complex conjugation.

See also

fr:Conjugu ja:共役複素数 pl:Liczba sprzężona sv:Komplexkonjugat sr:Коњуговано комплексни број

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