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Show that \begin{equation} \rho(\mathbf{x},\mathbf{y}) = \sqrt{\sum\limits_{i,j=1}^n a_{ij} (x_i-y_i)(x_j-y_j)}, \end{equation} with $a_{ij} = a_{ji}$, $\mathbf{x} = (x_1,\ldots \;, x_n)$ and $\mathbf{y} = (y_1,\ldots \;,y_n)$, satisfies the triangle inequality.

$[a_{ij}]_{n \times n}$, by the way, is a symmetric, positive-definite matrix with real entries.

I'm often pretty terrible at proving the triangle inequality, so any insights would be greatly appreciated.

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    try to prove that $ := x^t [a_{ij}] y$ is an inner product2011-03-28

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