A matrix is <non-normal matrix>[non-normal] when it does not commute with its <adjoint matrix>:
$$
LL^\dagger\ne L^\dagger L.
$$
Nonorthogonal decaying eigenmodes can interfere constructively and produce <transient growth>. For example,
$$
L=\begin{pmatrix}-1&10\\0&-2\end{pmatrix}
$$
has two negative <eigenvalues> but is non-normal. Starting from $x(0)=(0,1)^T$ gives
$$
x(t)=\begin{pmatrix}10(e^{-t}-e^{-2t})\\e^{-2t}\end{pmatrix}.
$$
At $t=\log2$, its squared <Euclidean norm> is $2.5^2+0.25^2>1=|x(0)|^2$. The energy grows transiently even though both eigenmodes eventually decay.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 331 / 3 / i / Solution

Transient growth is finite-time amplification in a stable linear system. For a non-normal generator it can occur even when every eigenvalue has negative real part, because nonorthogonal eigenmodes can interfere constructively.


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