Complex Conjugate and Norm

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Section 2.3 Complex Conjugate and Norm

Definition 2.3. Complex Conjugate.

The complex conjugate \(z^*\) of a complex number \(z=x+iy\) is found by replacing every \(i\) by \(-i\text{.}\) Therefore \(z^*=x-iy\text{.}\) (A common alternate notation for \(z^*\) is \(\bar{z}\text{.}\))

Geometrically, you should be able to see that the complex conjugate of any complex number is found by reflecting in the real axis, as shown in Figure 2.4.

Figure 2.4. The point \(z=x+iy\) and its complex conjugate \(z^*=x-iy\text{.}\) Note that, in this example, we have deliberately chosen \(z\) to be in the fourth quadrant to make you think about which of the variables \(x\text{,}\) \(y\text{,}\) and \(\phi\) are positive or negative.

You can find the conjugate of any complicated algebraic expression by taking the conjugate of all of the individual pieces. For example, for a matrix \(M\text{,}\) the conjugate \(M^*\) is found by taking the conjugate of each of the components.

Now let’s calculate an important product

\begin{align} z z^*\amp = (x+iy)(x-iy)\notag\\ \amp = x^2+y^2\notag\\ \amp = \vert z \vert^2\tag{2.3.1} \end{align}

Definition 2.5. Norm (or Magnitude) of a Complex Number.

Notice that the product \(zz^*\) is always a positive, real number. The (positive) square root of this number is the distance of the point \(z\) from the origin in the complex plane. We call the square root the norm or magnitude of \(z\) and we use the same notation as “absolute value,” i.e. \(\vert z\vert\text{.}\) In this way, we see that the definition of absolute value, as in \(\vert -2\vert=2\text{,}\) was never “strip off the minus sign,” but really “how far is \(-2\) from the origin.”