= Solution
If one retains the <Euler method>[Euler approximation] $A=I+J\delta t$, its exact <norm> <matrix> is
$$
A^TA=I+(J+J^T)\delta t+J^TJ\delta t^2.
$$
The additional term can mix the instantaneous <Symmetric matrix>[symmetric-part] <Eigenvector>[eigenvectors], so they no longer generally diagonalize finite-time error amplification. However, when $\delta t^2$ is material, a reliable physical <Fundamental matrix>[propagator] must itself be calculated beyond first-order Euler: a large <Euler method>[Euler-step] amplification can be a numerical artifact. For constant $J$, the exact map is $\Phi=e^{J\Delta t}$; for a changing reference trajectory it solves $\dot\Phi=J(t)\Phi$, $\Phi(0)=I$.
Use its <singular value decomposition>, $\Phi=U\Sigma V^T$. The initial orthonormal <Right singular vector>[right singular vectors] $\mathbf v_i$ diagonalize $\Phi^T\Phi$ and satisfy
$$
\boxed{\Phi\mathbf v_i=s_i\mathbf u_i,\qquad
\frac{\|\Phi\mathbf v_i\|^2}{\|\mathbf v_i\|^2}=s_i^2.}
$$
A <singular value> $s_i>1$ gives finite-time <norm> growth, $s_i<1$ decay, and the leading <right singular vector> is the initial perturbation with greatest amplification in the stated <norm>. The <left singular vector> describes the resulting final error pattern. <Non-normal matrix>[Non-normal] dynamics can give transient growth even if every <Jacobian matrix> <eigenvalue> has negative real part, so selecting observations from those <Eigenvalue>[eigenvalues] alone is insufficient. Over a genuinely short interval, $s_i^2=1+\lambda_i\delta t+O(\delta t^2)$, recovering the <Symmetric matrix>[symmetric-part] result. Initial observations should target the relevant growing right singular projections, with <Covariance matrix>[prior covariance] and observational precision included when they are not isotropic.
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