= Solution
The <induced matrix norm> associated with the given <energy norm> is a supremum of length ratios. The invertible <change of variables> $y=A^{1/2}x$ gives
$$
\|B\|_A=\sup_{x\ne0}\frac{\|Bx\|_A}{\|x\|_A}
=\sup_{y\ne0}\frac{\|A^{1/2}BA^{-1/2}y\|_2}{\|y\|_2}
=\boxed{\|A^{1/2}BA^{-1/2}\|_2}.
$$
The last expression is the Euclidean <operator norm>. This is the <matrix norm induced by an energy norm>: transforming to coordinates with Euclidean lengths transforms the operator by the corresponding <matrix similarity>.
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