What can we conclude about the norm of a complex vector (dimension 2) when it's equal to 0? What does it mean? What can be said about their components x and y?
That is not the norm of a 2-vector over $\displaystyle \mathbb{C}$. Assuming $\displaystyle \bold{x}$ is a column vector the usual norm is:
$\displaystyle \|\bold{x}\|=\sqrt{\overline{\bold{x}}^T{\bold{x}} }=\sqrt{x_1\overline{x_1}+x_2\overline{x_2}} =\sqrt{|x_1|^2+|x_2|^2}$
which $\displaystyle 0$ if and only if both $\displaystyle x_1$ and $\displaystyle x_2 =0$
CB
x=(5+2i,-2+5i)
Well,
||x||^2 = (5+2i)^2 + (-2+5i)^2
||x||^2 = (25 + 10i + 4i^2) + ( 4 -10i + 25i^2)
||x||^2 = (25+10i-4) + (4-10i-25) if we let (i^2 = -1)
||x||^2 = 25 + 10i -4 + 4 - 10i - 25
||x||^2 = 0
||x|| = 0
? Is there any mistake in my calculations? The initial components of x were not 0, yet the norm is zero.
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