What is the length of the vector (1,i) and why can it still be normalized?

[DocZaius]

I have a vector (1,i) and need to normalize it. I am being told that the answer is 1/(sqrt(2)) (1,i) but it seems clear to me that the vector's length is 0 and thus can't be normalized. What am i missing?

 

[I like Serena]

I have a vector (1,i) and need to normalize it. I am being told that the answer is 1/(sqrt(2)) (1,i) but it seems clear to me that the vector's length is 0 and thus can't be normalized. What am i missing?

Hi DocZaius!

The length of a vector containing complex numbers is defined slightly different.

From http://en.wikipedia.org/wiki/Norm_(mathematics):
On an n-dimensional complex space ℂⁿ the most common norm is
$$\|\boldsymbol{z}\| := \sqrt{|z_1|^2 + \cdots + |z_n|^2}= \sqrt{z_1 \bar z_1 + \cdots + z_n \bar z_n}.$$

 

[DocZaius]

Thanks!

 

[Fredrik]

The absolute value |z| of a complex number z=a+ib is defined by $|z|=\sqrt{a^2+b^2}$, or equivalently, by $|z|=\sqrt{z^*z}$, where z*=a-ib is the complex conjugate of z.