= Solution
For $q(t)=Bq_0$, the energy ratio is the <Rayleigh quotient> of the <Hermitian positive-definite matrix> $B^\dagger B$:
$$
\frac{\|Bq_0\|_2^2}{\|q_0\|_2^2}=\frac{q_0^\dagger B^\dagger Bq_0}{q_0^\dagger q_0}.
$$
Expanding $q_0$ in the orthonormal <right singular vectors> makes this a weighted average of $\sigma_j^2$. Hence
$$
\boxed{G(t)=\|e^{Lt}\|_2^2=\sigma_1^2(t),\qquad q_0\propto v_1(t).}
$$
Any nonzero vector in the largest right-singular subspace is optimal if $\sigma_1$ is repeated. Its amplified state is proportional to the paired <left singular vector>. For a <non-normal matrix> $L$, this optimal initial direction need not be an eigenvector of $L$, explaining <transient growth from non-normal modes>.
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