Proving isometry and continuity from a positive definite symmetric real matrix The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated.
Let $\epsilon$ be the Euclidean metric on $\Bbb R^n$ and let $A=[a_{ij}]_{n\times n}$ be a positive definite symmetric real matrix. Define
$$
\rho: \Bbb R^n\times\Bbb R^n\to\Bbb R_0^+,\quad(\mathbf x,\mathbf y)\mapsto \sqrt{\sum_{i,j=0}^{n}{a_{ij}(x_i-y_i)(x_j-y_j)}}
$$
where $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$


*

*Show that $(\Bbb R^n,\rho)$ and $(\Bbb R^n,\epsilon)$ are isometric
spaces.

*Show that the function $F: (\Bbb R^n,\epsilon)\to (\Bbb R^n,\rho),\quad x\mapsto x$ is continuous. 
 A: I'm not sure how are you supposed to do this, or what tools can you use. But linear algebra does wonders here:
Consider the vectors $\mathbf{x} $ and $\mathbf{y}$ as column matrices. Then one can write $\rho$ like this
$$\rho(\mathbf{x},\mathbf{y})=\sqrt{(\mathbf{x}-\mathbf{y})^tA(\mathbf{x}-\mathbf{y})}. $$ With this shorter notation, we can see where it actually comes from. Given a positive definite symmetric matrix $A$, one can define an inner product: $$\langle \mathbf{x},\mathbf{y}\rangle_A=\mathbf{x}^tA\mathbf{y}.$$ And given an inner product, one defines a norm $||\mathbf{x} ||_A=\sqrt{\langle\mathbf{x},\mathbf{x}\rangle_A}$. And finally, with the norm with define a metric $d_A(\mathbf{x},\mathbf{y})=||\mathbf{x}-\mathbf{y}||_A$. You can check that $\rho=d_A$. The Euclidian metric can also be seen in this point of view: $\epsilon=d_{Id}$.
Now all you need to do is find a (linear) map preserving the respective inner products, that is: $$\varphi:\mathbb{R}^n\to \mathbb{R}^n\ \mbox{such that}\ \langle\mathbf{x},\mathbf{y}\rangle_A=\langle\varphi(\mathbf{x}),\varphi(\mathbf{y})\rangle_{Id}$$ then the respective metrics will also be preserved.
For part $2$ you can use the linear map constructed above to show that the identity $\mathbf{x} \mapsto \mathbf{x}$ is actually Lipschtz.
