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Encyclopedia > Complex conjugate

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 z = a + ib (where a and b are real numbers) is defined to be z * = aib. The complex conjugate of a number z can be denoted by: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles â€” A collection of articles on various math topics, with interactive Java... In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (âˆ’1), which cannot be represented by any real number. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

$z^*_{}$ or $overline{z},!$

The symbol $A^* ,!$ can also denote the conjugate transpose of a matrix A so care must be taken not to confuse notations. If a complex number is treated as a 1×1 vector, the notations are identical. In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...

For example, (3 − 2i) * = 3 + 2i, i * = − i and 7 * = 7.

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. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

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

(z + w) * = z * + w *
(zw) * = z * w *
$left({frac{z}{w}}right)^* = frac{z^*}{w^*}$ if w is non-zero
z * = z if and only if z is real
$left| z^* right| = left| z right|$
${left| z right|}^2 = zz^*$
$z^{-1} = frac{z^*}{{left| z right|}^2}$ 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 p(z) = 0, then p(z * ) = 0 as well. Thus the roots of real polynomials outside of the real line occur in complex conjugate pairs. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

The function φ(z) = z * 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. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ...

## 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. In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... C*-algebras are an important area of research in functional analysis. ...

One may also define a conjugation for quaternions: the conjugate of a + bi + cj + dk is abicjdk. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

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

${left(zwright)}^* = w^* z^*.$

Since the multiplication of complex numbers is commutative, this reversal is not needed there. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...

There is also an abstract notion of conjugation for vector spaces V over the complex numbers. In this context, any (real) linear transformation $phi: V rightarrow V$ that satisfies The fundamental concept in linear algebra is that of a vector space or linear space. ... In mathematics, a linear transformation (also called linear operator <<wrong! operators are LTs on the same vector space or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

1. $phineq id_V$, the identity function on V,
2. φ2 = idV, and
3. φ(zv) = z * φ(v) for all $vin V$, $zin{mathbb C}$,

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. Canonical is an adjective derived from canon. ...

Results from FactBites:

 PlanetMath: complex conjugate (147 words) Complex conjugation represents a reflection about the real axis on the Argand diagram representing a complex number. Hence, the matrix complex conjugate is what we would expect: the same matrix with all of its scalar components conjugated. This is version 7 of complex conjugate, born on 2002-01-21, modified 2004-02-25.
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