calculating norm for a vector with complex element on it

Question

I would really appreciate it if someone can help me with complex numbers. I am haven't worked on it since graduation and decided to study again. I know this is pretty basic but I feel stuck on it.

I got the following vector below and I need to find the norm

$v = \begin{pmatrix} a \\ ai \end{pmatrix}$

My work as follows:

$\lvert\lvert v \lvert \lvert = \sqrt{a^2 + (ai)^2}$

$\lvert\lvert v \lvert \lvert = \sqrt{a^2 + a^2 * (i)^2}$

$\lvert\lvert v \lvert \lvert = \sqrt{a^2 + a^2 * (-1)}$

$\lvert\lvert v \lvert \lvert = \sqrt{a^2 - a^2}$

$\lvert\lvert v \lvert \lvert = 0$

But the answer for the solution is $$\lvert\lvert v \lvert \lvert = \sqrt{2}a$$

Can someone please point out the part that I am doing wrong here?

Thanks in advance

Answer 1

The formula for the norm you've used is only for real vectors. When your vector is complex it's

$$ \left\| { \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}}\right\|= \sqrt{|z_1|^2 + |z_2|^2}$$

where those $|\cdot|$ mean "magnitude of a complex number, i.e.

$$|x + iy| = \sqrt{x^2 + y^2}$$

If $z = x +iy$ this can also be written $$ |z| = \sqrt{z \bar z}$$

where $\bar z$ is the complex conjugate of $z$.

So the place where you went wrong is when you wrote

$$ a^2 +(ai)^2$$

It should have been

$$ |a|^2 +|ai|^2$$

This actually makes no difference in the first term, but it does in the second.

Also note that how you write this up depends on whether $a$ is a real number or a complex number. The answer you give from the solution actually only works if $a$ is a positive real number, so I guess you're supposed to assume that.

Answer 2

The standard sesqulinear form on $\mathbb{C}^n$ is defined as $$\langle x,y\rangle=\sum_{j=1}^nx_j\overline{y_j}$$ for vectors $x=(x_1,...,x_n)^{tr},y=(y_1,...,y_n)^{tr}\in\mathbb{C}^n$, where $x\mapsto\overline{x}$ is the complex conjugate. The norm is the squareroot of the quadratic form of this sesquilinear form: $$||x||=\sqrt{\langle x,x\rangle}=\sqrt{\sum_{j=1}^nx_j\overline{x_j}}=\sqrt{\sum_{j=1}^n|x_j|^2}=\sqrt{\sum_{j=1}^n\Re(x_j)^2+\Im(x_j)^2}.$$