共轭复数

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在數學上，共軛複數的定義如下：兩個複數若其實部相等，虛部互為相反數，則兩數互為共軛複數。複數 $z = a + i b$ (其中 $a$ , $b$ 均為實數) 的共軛複數如下：

$z^* \!\$ or $\overline{z} = a - ib \!\$

符號 $A *$ 也可表示矩陣 $A$ 的共軛轉置矩陣，因此需小心避免混淆。若複數表示為 $1\times 1$ 的向量，其表示方式不變。

例如 $(3 - 2 i) * = 3 + 2 i$ , $i * = - i$ , $7 * = 7$

一般常會將複數視為在直角坐標系的複數平面上的點。其 $x$ 座標為其實部，而 $y$ 座標為虛部。若以此觀點來看，共軛複數可視為一複數對應 X 軸的對應點。

若以極座標表示， $r e i φ$ 的共軛複數為 $r e - i φ$ 。這可以用歐拉公式來驗證。

[编辑] 属性

以下的性質對任意複數 $z$ 及 $w$ 都成立：（有特別說明者例外）

$(z + w)^* = z^* + w^* \!\$

$(zw)^* = z^* w^* \!\$

$\left({\frac{z}{w}}\right)^* = \frac{z^*}{w^*}$ 若

w

不為零

$z^* = z \!\$ 若且唯若

z

為實數

$\left| z^* \right| = \left| z \right|$

${\left| z \right|}^2 = zz^*$

$z^{-1} = \frac{z^*}{{\left| z \right|}^2}$ 若

z

不為零

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 non-real roots of real polynomials occur in complex conjugate pairs.

The function $φ(z) = z *$ from $\mathbf{C}$ to $\mathbf{C}$ 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 $\mathbf{C}/\mathbf{R}$ . This Galois group has only two elements: $φ$ and the identity on $\mathbf{C}$ . Thus the only two field automorphisms of $\mathbf{C}$ that leave the real numbers fixed are the identity map and complex conjugation.

[编辑] 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 $a + b i + c j + d k$ is $a - b i - c j - d k$ .

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

${\left(zw\right)}^* = w^* z^*.$

Since the multiplication of complex numbers is commutative, this reversal is not needed there.

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

$\phi\neq id_V$ , the identity function on $V$ ,
$φ 2 = i d V$ , and
$φ(z v) = z * φ(v)$ for all $v\in V$ , $z\in{\mathbb C}$ ,

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.