Definition

Scalar Complex Conjugate

Let $z$ be a complex number with real part $a$ and imaginary part $b$ ,

$$ z = a+bi $$

Then the complex conjugate of $z$ is

$$ \bar{z} = a - bi $$

Complex conjugation represents a reflection about the real axis on the Argand diagram representing a complex number.

Sometimes a star ($*$ ) is used instead of an overline, e.g. in physics you might see

$$ \int_{-\infty}^{\infty}\Psi^{*}\Psi dx=1 $$

where $\Psi^*$ is the complex conjugate of a wave function.

Matrix Complex Conjugate

Let $A=(a_{ij})$ be a $n\times m$ matrix with complex entries. Then the complex conjugate of $A$ is the matrix $\ccj{A}=(\ccj{a_{ij}})$ . In particular, if $v=(v^1, \ldots, v^n)$ is a complex row/column vector, then $\ccj{v}=(\ccj{v^1}, \ldots, \ccj{v^n})$ .

Hence, the matrix complex conjugate is what we would expect: the same matrix with all of its scalar components conjugated.

Properties of the Complex Conjugate

Scalar Properties

If $u,v$ are complex numbers, then

1. $\ccj{uv}= (\ccj{u})(\ccj{v})$
2. $\ccj{u+v}= \ccj{u}+\ccj{v}$
3. $\big(\ccj{u}\big)^{-1} = \ccj{u^{-1}}$
4. $\ccj{(\ccj{u})} = u$
5. If $v\neq 0$ , then $\ccj{(\frac{u}{v})} = {\ccj{u}}/{\ccj{v}}$
6. Let $u = a + bi$ . Then $\ccj{u} u = u \ccj{u} = a^2+b^2 \ge 0$ (the complex modulus).
7. If $z$ is written in polar form as $z=r e^{i\phi}$ , then $\ccj{z}=re^{-i\phi}$ .

Matrix and Vector Properties

Let $A$ be a matrix with complex entries, and let $v$ be a complex row/column vector.

Then

1. $\ccj{A^T}=\big(\ccj{A}\big)^T$
2. $\ccj{Av}=\ccj{A}\ccj{v}$ , and $\ccj{vA}=\ccj{v}\ccj{A}$ . (Here we assume that $A$ and $v$ are compatible size.)

Now assume further that $A$ is a complex square matrix, then

1. $\trace \ccj{A} = \ccj{(\trace\ A)}$
2. $\det \ccj{A} = \ccj{(\det A)}$
3. $\big(\ccj{A}\big)^{-1} = \ccj{A^{-1}}$