Reasoning about Equality of norms

Question

This might seem absurd, but I am a bit confused! I have the following norm equation $\Vert x + y \Vert_2 = \Vert x \Vert_2$.

Can one conclude that $y = 0$? any solution?

Answer 1

You can not conclude $y=0$ ! For instance, in $\mathbb{R}$, you have $|1-2|=|-1|=|1|=1$ but $2\not = 0$.

(Note that in $\mathbb{R}$, $\Vert x\Vert_2 = \sqrt{x^2}=|x|$.)

Answer 2

You haven't said what space $x$ and $y$ live in, so I'm going to assume a relatively simple case, that it's $\mathbb R^2$. Let $x = (x_1,x_2)$ and $y=(y_1,y_2)$, then the given equation becomes $$ \sqrt{(x_1+y_1)^2+(x_2+y_2)^2} = \sqrt{x_1^2+x_2^2}$$ Squaring both sides and canceling some terms gives $$ 2x_1y_1 + y_1^2 + 2x_2y_2 + y_2^2 = 0 $$ which has more solutions than $y=0$. E.g., consider $x=(1,1)$ and $y=(0,-2)$.