linear algebra norm from inner space

Question

so we have a symmetric diagonal matrix W and I have proven that the following is an inner product $$\langle x,y\rangle_W = x^TWy$$ now I am asked to prove that $$\|x\| = \langle x, x \rangle$$ is a norm. is there a mistake in the question? isn't the correct norm with a square root like this $$\|x\| = \sqrt{\langle x, x \rangle }$$ or I am I missing something?

Answer 1

The second one is, in general, the norm inducted by an inner product. But you can ask if the first is or no a norm. The second one is called so because if you have a norm for which is true the parallelogram's identity it's defined an inner product such that the $\sqrt{<.,.>}$ is the originary norm. (I didn't check ii the first is an inner product, if you had a general matrix or a specific one).

Answer 2

For $x \neq 0$, $\langle x, x\rangle_W>0$. Take $x=\begin{pmatrix} a \\ a \end{pmatrix}$, $W=\begin{pmatrix} a & 0\\ 0 & -a \end{pmatrix}$ for $a\neq0$. Then calculate that $\langle x, x\rangle_W=x^TWx=0$ but $x\neq0$. That's not an inner product.