SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\Re (\lambda_ {r} (\mathbf {W}) \mathrm {f} _ {\alpha} (\lambda_ {l} (\mathbf {L}))) = \lambda_ {r} (\mathbf {W}) | \lambda (\mathbf {L}) _ {i} | ^ {\alpha} \cos (\theta_ {i}) = \lambda_ {r} (\mathbf {W}) | \lambda (\mathbf {L}) _ {i} | ^ {\alpha - 1} \Re (\lambda (\mathbf {L}) _ {i}).
$$
By Perron-Frobenius, the eigenvalue of  $\mathbf{L}$  with the largest eigenvalues is given by  $\lambda_N(\mathbf{L})\in \mathbb{R}$ . Hence, for all  $l = 1,\dots ,N$
$$
\left| \lambda (\mathbf {L}) _ {l} \right| ^ {\alpha} \cos (\theta_ {l}) \leq \left| \lambda (\mathbf {L}) _ {N} \right| ^ {\alpha}.
$$
Similarly, for all  $l = 1,\dots ,N$  with  $\Re (\lambda (\mathbf{L})_l) <   0$
$$
- \left| \lambda (\mathbf {L}) _ {l} \right| ^ {\alpha} \cos \left(\theta_ {l}\right) \leq - \left| \lambda (\mathbf {L}) _ {1} \right| ^ {\alpha}.
$$
Thus, the frequency with the most negative real part is either given by  $\lambda_K(\mathbf{W})\mathrm{f}_{\alpha}(\lambda_1(\mathbf{L}))$  or  $\lambda_{1}(\mathbf{W})\mathrm{f}_{\alpha}(\lambda_{N}(\mathbf{L}))$ . The remainder of the proof is analogous to the proof of Theorem D.7.
In the following, we provide the complete statement and proof for the claims made in Theorem 5.3 when the underlying ODE is the Schrödinger equation as presented in (3).
Theorem D.8. Let  $\mathcal{G}$  be a undirected graph with SNA L. Consider the initial value problem in (3) with channel mixing matrix  $\mathbf{W} \in \mathbb{C}^{K \times K}$  and  $\alpha \in \mathbb{R}$ . Suppose that  $\mathbf{W}$  has at least one eigenvalue with non-zero imaginary part and sort the eigenvalues of  $\mathbf{W}$  in ascending order with respect to their imaginary part. Then, for almost initial values  $\mathbf{x}_0 \in \mathbb{C}^{N \times K}$ , the following is satisfied.
$(\alpha >0)$  Solutions of (3) are HFD if
$$
\Im \left(\lambda_ {K} (\mathbf {W})\right) f _ {\alpha} \left(\lambda_ {1} (\mathbf {L})\right) <   \Im \left(\lambda_ {1} (\mathbf {W})\right),
$$
$(\alpha < 0)$  Let  $\lambda_{+}(\mathbf{L})$  and  $\lambda_{-}(\mathbf{L})$  be the smallest positive and biggest negative non-zero eigenvalue of  $\mathbf{L}$ , respectively. Solutions of (3) are  $(1 - \lambda_{-}(\mathbf{L}))$ -FD if
$$
\Im \left(\lambda_ {K} (\mathbf {W})\right) f _ {\alpha} (\lambda_ {-} (\mathbf {L})) <   \Im \left(\lambda_ {1} (\mathbf {W})\right) f _ {\alpha} (\lambda_ {+} (\mathbf {L})).
$$
Otherwise, solutions of (3) are  $(1 - \lambda_{+}(\mathbf{L}))$ -FD.
Proof. The proof follows the same reasoning as the proof for the heat equation in Theorem D.5. The difference is that we now apply Lemma D.4 instead of Lemma D.3.
Therefore, the dominating frequency is either  $\lambda_K(\mathbf{W})\mathrm{f}_{\alpha}(\lambda_1(\mathbf{L}))$  or  $\lambda_{1}(\mathbf{W})\mathrm{f}_{\alpha}(\lambda_{N}(\mathbf{L}))$  if  $\alpha >0$  and  $\lambda_{K}(\mathbf{W})\mathrm{f}_{\alpha}\left(\lambda_{-}(\mathbf{L})\right)$  or  $\lambda_{1}(\mathbf{W})\mathrm{f}_{\alpha}\left(\lambda_{+}(\mathbf{L})\right)$  if  $\alpha < 0$
D.2 Frequency Dominance for Numerical Approximations of the Heat Equation
For  $n \in \mathbb{N}$  and  $h \in \mathbb{R}$ ,  $h > 0$ , the solution of (2) at time  $nh > 0$  can be approximated with an explicit Euler scheme
$$
\operatorname {v e c} (\mathbf {x}) (n   h) = \sum_ {k = 0} ^ {n} {\binom {n} {k}} h ^ {k} (- \mathbf {W} \otimes \mathbf {L} ^ {\alpha}) ^ {k} \operatorname {v e c} (\mathbf {x} _ {0}),
$$
which can be further simplified via the binomial theorem as
$$
\operatorname {v e c} (\mathbf {x}) (n h) = \left(\mathbf {I} - h \left(\mathbf {W} \otimes \mathbf {L} ^ {\alpha}\right)\right) ^ {n} \operatorname {v e c} \left(\mathbf {x} _ {0}\right). \tag {11}
$$
Hence, it holds the representation formula
$$
\operatorname {v e c} (\mathbf {x}) (n h) = \sum_ {r, l} c _ {r, l} \left(1 - h \lambda_ {r} (\mathbf {W}) \mathrm {f} _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right)\right) ^ {n} \psi_ {r} (\mathbf {W}) \otimes \psi_ {l} (\mathbf {L}).
$$
In this case, the dominating frequency maximizes  $|1 - h\lambda_r(\mathbf{W})\mathrm{f}_\alpha (\lambda_l(\mathbf{L}))|$ . When  $h < \| \mathbf{W}\|^{-1}$ , the product  $h\lambda_r(\mathbf{W})\mathrm{f}_\alpha (\lambda_l(\mathbf{L}))$  is guaranteed to be in  $[-1,1]$ , and
$$
\left. \left| 1 - h \lambda_ {r} (\mathbf {W}) f _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right) \right| = 1 - h \lambda_ {r} (\mathbf {W}) f _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right) \in [ 0, 2 ] \right..
$$
Therefore, the dominating frequency minimizes  $h\lambda_r(\mathbf{W})\mathrm{f}_\alpha (\lambda_l(\mathbf{L}))$ . This is the reasoning behind the next result.
Proposition D.9. Let  $h \in \mathbb{R}$ ,  $h > 0$ . Consider the fractional heat equation (2) with  $\alpha \in \mathbb{R}$ . Let  $\{\mathbf{x}(nh)\}_{n \in \mathbb{N}}$  be the trajectory of vectors derived by approximating (2) with an explicit Euler scheme with step size  $h$ . Suppose  $h < \| \mathbf{W} \|^{-1}$ . Then, for almost all initial values  $\mathbf{x}_0$
$$
\mathscr {E} \left(\frac {\mathbf {x} (n   h)}{\| \mathbf {x} (n   h) \| _ {2}}\right) \xrightarrow {n \to \infty} \left\{ \begin{array}{l l} \frac {\lambda_ {N} (\mathbf {I} - \mathbf {L})}{2}, & i f \lambda_ {K} (\mathbf {W})   \mathrm {f} _ {\alpha} \left(\lambda_ {1} (\mathbf {L})\right) <   \lambda_ {1} (\mathbf {W}), \\ 0  , & o t h e r w i s e  . \end{array} \right.
$$
Proof. Define
$$
\left(\lambda_ {a}, \lambda_ {b}\right) := \underset {r, l} {\arg \max } \left\{\left| 1 - h \lambda_ {r} \left(\mathbf {W}\right) \mathrm {f} _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right) \right|: r \in \{1, \ldots , K \}, l \in \{1, \ldots , N \} \right\}.
$$
By the hypothesis on  $h$ , this is equivalent to
$$
\left(\lambda_ {a}, \lambda_ {b}\right) = \underset {r, l} {\arg \min } \left\{\lambda_ {r} \left(\mathbf {W}\right) \mathrm {f} _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right): r \in \{1, \dots , K \}, l \in \{1, \dots , N \} \right\}.
$$
Therefore,  $(\lambda_{a},\lambda_{b})$  is either  $(\lambda_{1}(\mathbf{W}),\lambda_{N}(\mathbf{L}))$  or  $(\lambda_K(\mathbf{W}),\lambda_1(\mathbf{L}))$ . Hence,
$$
\frac {\operatorname {v e c} (\mathbf {x}) (n h)}{\| \operatorname {v e c} (\mathbf {x}) (n h) \| _ {2}} \xrightarrow {n \to \infty} \frac {c _ {a , b} \psi_ {a} (\mathbf {W}) \otimes \psi_ {b} (\mathbf {L})}{\| c _ {a , b} \psi_ {a} (\mathbf {W}) \otimes \psi_ {b} (\mathbf {L}) \| _ {2}}.
$$
If the condition  $\lambda_K(\mathbf{W})\mathrm{f}_{\alpha}(\lambda_1(\mathbf{L})) < \lambda_1(\mathbf{W})$  is satisfied, we have  $b = 1$ . Then by (10), the normalized  $\operatorname{vec}(\mathbf{x})$  converges to the eigenvector of  $\mathbf{I}\otimes \mathbf{I} - \mathbf{L}$  corresponding to the largest frequency  $1 - \lambda_{1}(\mathbf{L}) = \lambda_{N}(\mathbf{I} - \mathbf{L})$ . An application of Lemma D.2 finishes the proof.
If  $\lambda_K(\mathbf{W})\mathrm{f}_\alpha (\lambda_1(\mathbf{L})) < \lambda_1(\mathbf{W})$  is not satisfied, we have  $b = N$ , and the other direction follows with the same argument.
Similarly to Proposition D.9 one can prove the following results for negative fractions.
Proposition D.10. Let  $h \in \mathbb{R}$ ,  $h > 0$ . Consider the fractional heat equation (2) with  $\alpha < 0$ . Let  $\{\mathbf{x}(nh)\}_{n \in \mathbb{N}}$  be the trajectory of vectors derived by approximating the solution of (2) with an explicit Euler scheme with step size  $h$ . Suppose that  $h < \| \mathbf{W} \|^{-1}$ . The approximated solution is  $(1 - \lambda_{-}(\mathbf{L}))$ -FD if
$$
\lambda_ {1} (\mathbf {W}) f _ {\alpha} \left(\lambda_ {+} (\mathbf {L})\right) <   \lambda_ {K} (\mathbf {W}) f _ {\alpha} \left(\lambda_ {-} (\mathbf {L})\right),
$$
Proof. The proof follows the same reasoning as the proof of Proposition D.9 by realizing that the dominating frequencies  $(\lambda_{a},\lambda_{b})$  are either given by  $(\lambda_{1}(\mathbf{W}),\lambda_{+}(\mathbf{L}))$  or  $(\lambda_K(\mathbf{W}),\lambda_{-}(\mathbf{L}))$ .
D.3 Frequency Dominance for Numerical Approximations of the Schrödinger Equation
For  $n \in \mathbb{N}$  and  $h \in \mathbb{R}$ ,  $h > 0$ , the solution of (3) at time  $nh > 0$  can be approximated with an explicit Euler scheme as well. Similarly to the previous section, we can write
$$
\operatorname {v e c} (\mathbf {x}) (n h) = \left(\mathbf {I} + i h \left(\mathbf {W} \otimes \mathbf {L} ^ {\alpha}\right)\right) ^ {n} \operatorname {v e c} (\mathbf {x} _ {0}).
$$
$$
\operatorname {v e c} (\mathbf {x}) (n h) = \sum_ {r, l} c _ {r, l} \left(1 + i h \lambda_ {r} (\mathbf {W}) \mathrm {f} _ {\alpha} (\lambda_ {l} (\mathbf {L}))\right) ^ {n} \psi_ {r} (\mathbf {W}) \otimes \psi_ {l} (\mathbf {L}).
$$
The dominating frequency will be discussed in the following theorem.
Proposition D.11. Let  $h \in \mathbb{R}$ ,  $h > 0$ . Let  $\{\mathbf{x}(nh)\}_{n \in \mathbb{N}}$  be the trajectory of vectors derived by approximating (3) with an explicit Euler scheme with sufficiently small step size  $h$ . Sort the eigenvalues of  $\mathbf{W}$  in ascending order with respect to their imaginary part. Then, for almost all initial values  $\mathbf{x}_0$
$$
\mathscr {E} \left(\frac {\mathbf {x} (n h)}{\| \mathbf {x} (n h) \| _ {2}}\right) \xrightarrow {n \to \infty} \left\{ \begin{array}{l l} \frac {\lambda_ {N} (\mathbf {I} - \mathbf {L})}{2}, & i f f _ {\alpha} \left(\lambda_ {1} (\mathbf {L})\right) \mathfrak {S} \left(\lambda_ {K} (\mathbf {W})\right) <   f _ {\alpha} \left(\lambda_ {N} (\mathbf {L})\right) \mathfrak {S} \left(\lambda_ {1} (\mathbf {W})\right) \\ 0, & o t h e r w i s e. \end{array} \right.
$$
$$
\left(\lambda_ {a}, \lambda_ {b}\right) := \underset {r, l} {\arg \max } \left\{\left| 1 + i   h \lambda_ {r} \left(\mathbf {W}\right) \mathrm {f} _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right) \right|: r \in \{1, \ldots , K \}  ,   l \in \{1, \ldots , N \} \right\}.
$$
By definition of  $a$  and  $b$ , for all  $r$  and  $l$  it holds
$$
\left| 1 + i h \lambda_ {a} (\mathbf {W}) f _ {\alpha} (\lambda_ {b} (\mathbf {L})) \right| > \left| 1 + i h \lambda_ {r} (\mathbf {W}) f _ {\alpha} (\lambda_ {l} (\mathbf {L})) \right|. \tag {12}
$$
$$
\frac {\operatorname {v e c} (\mathbf {x}) (t)}{\| \operatorname {v e c} (\mathbf {x}) (t) \| _ {2}} \xrightarrow {t \to \infty} \frac {c _ {a , b} \psi_ {a} (\mathbf {W}) \otimes \psi_ {b} (\mathbf {L})}{\| c _ {a , b} \psi_ {a} (\mathbf {W}) \otimes \psi_ {b} (\mathbf {L}) \| _ {2}}.
$$
We continue by determining the indices  $a$  and  $b$ . To do so, we note that (12) is equivalent to
$$
\begin{array}{l} \mathrm {f} _ {\alpha} (\lambda_ {l} (\mathbf {L})) \Im (\lambda_ {r} (\mathbf {W})) - \mathrm {f} _ {\alpha} (\lambda_ {b} (\mathbf {L})) \Im (\lambda_ {a} (\mathbf {W})) \\ > \frac {h}{2} \left(\mathrm {f} _ {\alpha} (\lambda_ {l} (\mathbf {L})) ^ {2} | \lambda_ {r} (\mathbf {W}) | ^ {2} - \mathrm {f} _ {\alpha} (\lambda_ {b} (\mathbf {L})) ^ {2} | \lambda_ {a} (\mathbf {W}) | ^ {2}\right) \\ \end{array}
$$
for all  $r, l$ . Denote by  $\varepsilon$  the gap
$$
0 <   \varepsilon := \min  _ {(r, l) \neq (a, b)} \left\{\mathrm {f} _ {\alpha} \left(\lambda_ {l} \left(\mathbf {L}\right)\right) \Im \left(\lambda_ {r} \left(\mathbf {W}\right)\right) - \mathrm {f} _ {\alpha} \left(\lambda_ {b} \left(\mathbf {L}\right)\right) \Im \left(\lambda_ {a} \left(\mathbf {W}\right)\right) \right\}.
$$
Noting that
$$
\left\{ \begin{array}{l} \frac {h}{2} \left(\mathrm {f} _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right) ^ {2} \left| \lambda_ {r} (\mathbf {W}) \right| ^ {2} - \mathrm {f} _ {\alpha} \left(\lambda_ {b} (\mathbf {L})\right) ^ {2} \left| \lambda_ {a} (\mathbf {W}) \right| ^ {2}\right) \leq h \| \mathbf {W} \| ^ {2} \| \mathbf {L} \| ^ {2 \alpha} = h \| \mathbf {W} \| ^ {2}, \\ \frac {h}{2} \left(\mathrm {f} _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right) ^ {2} \left| \lambda_ {r} (\mathbf {W}) \right| ^ {2} - \mathrm {f} _ {\alpha} \left(\lambda_ {b} (\mathbf {L})\right) ^ {2} \left| \lambda_ {u} (\mathbf {W}) \right| ^ {2}\right) <   \varepsilon \end{array} \right.
$$
one gets that (12) is satisfied for  $h < \varepsilon \| \mathbf{W} \|^ {- 2}$ . Therefore, for sufficiently small  $h$ , the dominating frequencies are the ones with minimal imaginary part, i.e., either  $\mathrm{f}_{\alpha}\left(\lambda_{1}(\mathbf{L})\right) \Im \left(\lambda_{K}(\mathbf{W})\right)$  or  $\mathrm{f}_{\alpha}\left(\lambda_{N}(\mathbf{L})\right) \Im \left(\lambda_{1}(\mathbf{W})\right)$ . If  $\mathrm{f}_{\alpha}\left(\lambda_{1}(\mathbf{L})\right) \Im \left(\lambda_{K}(\mathbf{W})\right) < \mathrm{f}_{\alpha}\left(\lambda_{N}(\mathbf{L})\right) \Im \left(\lambda_{1}(\mathbf{W})\right)$ , then  $b = 1$ , and the normalized vec  $(\mathbf{x})$  converges to the eigenvector corresponding to the smallest frequency  $\lambda_{1}(\mathbf{L})$ . By (10), this is also the eigenvector of  $\mathbf{I} \otimes \mathbf{I} - \mathbf{L}$  corresponding to the largest frequency  $1 - \lambda_{1}(\mathbf{L}) = \lambda_{N}(\mathbf{I} - \mathbf{L})$ . An application of Lemma D.2 finishes the proof.
Finally, we present a similar result for negative powers.
Proposition D.12. Let  $h \in \mathbb{R}$ ,  $h > 0$ . Consider the fractional Schrödinger equation (3) with  $\alpha < 0$ . Let  $\{\mathbf{x}(nh)\}_{n \in \mathbb{N}}$  be the trajectory of vectors derived by approximating the solution of (3) with an explicit Euler scheme with step size  $h$ . Suppose that  $h$  is sufficiently small. Sort the eigenvalues of  $\mathbf{W}$  in ascending order with respect to their imaginary part. The approximated solution is  $(1 - \lambda_{+}(\mathbf{L}))$ -FD if
$$
\lambda_ {1} (\mathbf {W}) f _ {\alpha} (\lambda_ {+} (\mathbf {L})) <   \lambda_ {K} (\mathbf {W}) f _ {\alpha} (\lambda_ {-} (\mathbf {L})) ,
$$
and  $(1 - \lambda_{-}(\mathbf{L}))$ -FD otherwise.
Proof. Similar to Proposition D.11, we can prove the statement by realizing that the dominating frequencies  $(\lambda_{a},\lambda_{b})$  in (12) are either given by  $(\lambda_{1}(\mathbf{W}),\lambda_{+}(\mathbf{L}))$  or  $(\lambda_K(\mathbf{W}),\lambda_-(\mathbf{L}))$ .
E Appendix for Section 5.2
We begin this section by describing the solution of general linear matrix ODEs of the form (6) in terms of the Jordan decomposition of  $\mathbf{M}$ . This is required when  $\mathbf{M}$  is not diagonalizable. For instance, the SNA of a directed graph is not in general a symmetric matrix, hence, not guaranteed to be diagonalizable. We then proceed in Appendix E.1 with the proof of Theorem 5.6.
For a given matrix  $\mathbf{M} \in \mathbb{C}^{N \times N}$ , the Jordan normal form is given by
$$
\mathbf {M} = \mathbf {P J P} ^ {- 1},
$$
where  $\mathbf{P} \in \mathbb{C}^{N \times N}$  is an invertible matrix whose columns are the generalized eigenvectors of  $\mathbf{M}$ , and  $\mathbf{J} \in \mathbb{C}^{N \times N}$  is a block-diagonal matrix with Jordan blocks along its diagonal. Denote with  $\lambda_1, \ldots, \lambda_m$  the eigenvalues of  $\mathbf{M}$  and with  $\mathbf{J}_1, \ldots, \mathbf{J}_m$  the corresponding Jordan blocks. Let  $k_l$  be the algebraic multiplicity of the eigenvalue  $\lambda_l$ , and denote with  $\{\psi_l^i(\mathbf{M})\}_{i \in \{1, \ldots, k_l\}}$  the generalized eigenvectors of the Jordan block  $\mathbf{J}_l$ .
We begin by giving the following well-known result, which fully characterizes the frequencies for the solution of a linear matrix ODE.
Lemma E.1. Let  $\mathbf{M} = \mathbf{P}\mathbf{J}\mathbf{P}^{-1} \in \mathbb{C}^{N \times N}$  be the Jordan normal form of  $\mathbf{M}$ . Let  $\mathbf{x}: [0, T] \to \mathbb{R}^n$  be a solution to
$$
\mathbf {x} ^ {\prime} (t) = \mathbf {M} \mathbf {x} (t), \mathbf {x} (0) = \mathbf {x} _ {0}.
$$
Then,  $\mathbf{x}$  is given by
$$
\mathbf {x} (t) = \sum_ {l = 1} ^ {m} \exp {(\lambda_ {l} (\mathbf {M}) t)} \sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {j} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j}}{(i - j) !} \psi_ {l} ^ {j} (\mathbf {M}),
$$
$$
\mathbf {x} _ {0} = \sum_ {l = 1} ^ {m} \sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {i} \mathbf {P e} _ {l} ^ {i},
$$
and  $\{\mathbf{e}_l^i : i \in \{1, \dots, k_l\}, l \in \{1, \dots, m\}\}$  is the standard basis satisfying  $\mathbf{P}\mathbf{e}_l^i = \psi_l^i(\mathbf{M})$ .
Proof. By (Perko, 2001, Section 1.8), the solution can be written as
$$
\exp \left(\mathbf {M} t\right) \mathbf {x} _ {0} = \mathbf {P} \exp \left(\mathbf {J} t\right) \mathbf {P} ^ {- 1} \left(\sum_ {l = 1} ^ {m} \sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {i} \mathbf {P e} _ {l} ^ {i}\right) = \mathbf {P} \exp \left(\mathbf {J} t\right) \left(\sum_ {l = 1} ^ {m} \sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {i} \mathbf {e} _ {l} ^ {i}\right),
$$
where  $\exp (\mathbf{J}t) = \mathrm{diag}\left(\{\exp (\mathbf{J}_l t)\}_{l = 1}^m\right)$  and
$$
\exp \left(\mathbf {J} _ {l} t\right) = \exp \left(\lambda_ {l} (\mathbf {M})   t\right) \left[ \begin{array}{c c c c c} 1 & t & \frac {t ^ {2}}{2 !} & \dots & \frac {t ^ {k _ {l}}}{(k _ {l} - 1) !} \\ & 1 & t & & \vdots \\ & & 1 & \ddots & \frac {t ^ {2}}{2 !} \\ & & & \ddots & t \\ & & & & 1 \end{array} \right].
$$
Since  $\exp (\mathbf{J}t) = \bigoplus_{l = 1}^{m}\exp (\mathbf{J}_{l}t)$ , we can focus on a single Jordan block. Fix  $l\in \{1,\ldots ,m\}$ , it holds
$$
\begin{array}{l} \mathbf {P} \exp \left(\mathbf {J} _ {l} t\right) \left(\sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {i} \mathbf {e} _ {l} ^ {i}\right) \\ = \mathbf {P} \exp (\lambda_ {l} (\mathbf {M}) t) \left(c _ {l} ^ {1} \mathbf {e} _ {l} ^ {1} + c _ {l} ^ {2} (t \mathbf {e} _ {l} ^ {1} + \mathbf {e} _ {l} ^ {2}) + c _ {l} ^ {3} \left(\frac {t ^ {2}}{2 !} \mathbf {e} _ {l} ^ {1} + t \mathbf {e} _ {l} ^ {2} + \mathbf {e} _ {l} ^ {3}\right) + \dots\right) \\ = \exp (\lambda_ {l} (\mathbf {M}) t) \left(c _ {l} ^ {1} \psi_ {l} ^ {1} (\mathbf {M}) + c _ {l} ^ {2} (t \psi_ {l} ^ {1} (\mathbf {M}) + \psi_ {l} ^ {2} (\mathbf {M})) \right. \\ \left. + c _ {l} ^ {3} \left(\frac {t ^ {2}}{2 !} \psi_ {l} ^ {1} (\mathbf {M}) + t \psi_ {l} ^ {2} (\mathbf {M}) + \psi_ {l} ^ {3} (\mathbf {M})\right) + \dots\right) \\ = \exp \left(\lambda_ {l} (\mathbf {M}) t\right) \sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j}}{(i - j) !} \psi_ {l} ^ {j} (\mathbf {M}). \\ \end{array}
$$
Bringing the direct sums together, we get
$$
\exp \left(\mathbf {M} t\right) \mathbf {x} _ {0} = \sum_ {l = 1} ^ {m} \exp \left(\lambda_ {l} (\mathbf {M}) t\right) \sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j}}{(i - j) !} \psi_ {l} ^ {j} (\mathbf {M}),
$$
In the following, we derive a formula for the solution of ODEs of the form
$$
\mathbf {x} ^ {\prime} (t) = \mathbf {M} \mathbf {x} (t) \mathbf {W}, \mathbf {x} (0) = \mathbf {x} _ {0}, \tag {13}
$$
for a diagonal matrix  $\mathbf{W} \in \mathbb{C}^{K \times K}$  and a general square matrix  $\mathbf{M} \in \mathbb{C}^{N \times N}$  with Jordan normal form  $\mathbf{PJP}^{-1}$ . By vectorizing, we obtain the equivalent linear system
$$
\operatorname {v e c} (\mathbf {x}) ^ {\prime} (t) = \mathbf {W} \otimes \mathbf {M} \operatorname {v e c} (\mathbf {x}) (t), \operatorname {v e c} (\mathbf {x}) (0) = \operatorname {v e c} \left(\mathbf {x} _ {0}\right). \tag {14}
$$
Then, by properties of the Kronecker product, there holds
$$
\mathbf {W} \otimes \mathbf {M} = \mathbf {W} \otimes (\mathbf {P J P} ^ {- 1}) = (\mathbf {I} \otimes \mathbf {P}) (\mathbf {W} \otimes \mathbf {J}) (\mathbf {I} \otimes \mathbf {P} ^ {- 1}) = (\mathbf {I} \otimes \mathbf {P}) (\mathbf {W} \otimes \mathbf {J}) (\mathbf {I} \otimes \mathbf {P}) ^ {- 1}.
$$
Note that  $(\mathbf{I}\otimes \mathbf{P})(\mathbf{W}\otimes \mathbf{J})(\mathbf{I}\otimes \mathbf{P})^{-1}$  is not the Jordan normal form of  $\mathbf{D}\otimes \mathbf{M}$ . However, we can characterize the Jordan form of  $\mathbf{W}\otimes \mathbf{M}$  as follows.
Lemma E.2. The Jordan decomposition of  $\mathbf{W} \otimes \mathbf{J}$  is given by  $\mathbf{W} \otimes \mathbf{J} = \tilde{\mathbf{P}}\tilde{\mathbf{J}}\tilde{\mathbf{P}}^{-1}$  where  $\tilde{\mathbf{J}}$  is a block diagonal matrix with blocks
$$
\tilde {\mathbf {J}} _ {j, l} = \left[ \begin{array}{c c c c c} w _ {j} \lambda_ {l} (\mathbf {J}) & 1 & & & \\ & w _ {j} \lambda_ {l} (\mathbf {J}) & 1 & & \\ & & \ddots & & \\ & & & w _ {j} \lambda_ {l} (\mathbf {J}) & 1 \\ & & & & w _ {j} \lambda_ {l} (\mathbf {J}) \end{array} \right],
$$
and  $\tilde{\mathbf{P}}$  is a diagonal matrix obtained by concatenating  $\tilde{\mathbf{P}}_{j,l} = \mathrm{diag}\left(\left\{w_j^{-n + 1}\right\}_{n = 1}^{k_l}\right)$ .
Proof. As  $\mathbf{J} = \bigoplus_{l=1}^{m} \mathbf{J}_{l}$ , we can focus on a single Jordan block. Fix  $l \in \{1, \ldots, m\}$ . We have
$$
\mathbf {W} \otimes \mathbf {J} _ {l} = \operatorname {d i a g} \left(\left\{w _ {j} \mathbf {J} _ {l} \right\} _ {j = 1} ^ {K}\right) = \bigoplus_ {j = 1} ^ {K} w _ {j} \mathbf {J} _ {l},
$$
hence, we can focus once again on a single block. Fix  $j \in \{1, \dots, K\}$ ; the Jordan decomposition of  $w_{j}\mathbf{J}_{l}$  is given by  $\tilde{\mathbf{P}}_l = \mathrm{diag}\left(\left\{w_j^{-n + 1}\right\}_{n = 1}^{k_l}\right)$  and
$$
\tilde {\mathbf {J}} _ {l} = \left[ \begin{array}{c c c c c c} w _ {j} \lambda_ {l} (\mathbf {J}) & 1 & & & & \\ & w _ {j} \lambda_ {l} (\mathbf {J}) & 1 & & & \\ & & \ddots & \ddots & & \\ & & & & w _ {j} \lambda_ {l} (\mathbf {J}) & 1 \\ & & & & & w _ {j} \lambda_ {l} (\mathbf {J}) \end{array} \right].
$$
To verify it, compute the  $(n,m)$  element
$$
\left(\tilde {\mathbf {P}} _ {l} \tilde {\mathbf {J}} _ {l} \tilde {\mathbf {P}} _ {l} ^ {- 1}\right) _ {n, m} = \sum_ {i, k} \left(\tilde {\mathbf {P}} _ {l}\right) _ {n, i} \left(\tilde {\mathbf {J}} _ {l}\right) _ {i, k} \left(\tilde {\mathbf {P}} _ {l} ^ {- 1}\right) _ {k, m}.
$$
Since  $\tilde{\mathbf{P}}_l$  is a diagonal matrix, the only non-null entries are on the diagonal; therefore,  $i = n$  and  $k = m$
$$
= \left(\tilde {\mathbf {P}} _ {l}\right) _ {n, n} \left(\tilde {\mathbf {J}} _ {l}\right) _ {n, m} \left(\tilde {\mathbf {P}} _ {l} ^ {- 1}\right) _ {m, m}
$$
and the only non-null entries of  $\tilde{\mathbf{J}}_l$  are when  $m = n$  or  $m = n + 1$ , hence
$$
= \left\{ \begin{array}{l l} \left(\tilde {\mathbf {P}} _ {l}\right) _ {n, n} \left(\tilde {\mathbf {J}} _ {l}\right) _ {n, n} \left(\tilde {\mathbf {P}} _ {l} ^ {- 1}\right) _ {n, n} = w _ {j} \lambda_ {l} (\mathbf {J})  , & m = n  , \\ \left(\tilde {\mathbf {P}} _ {l}\right) _ {n, n} \left(\tilde {\mathbf {J}} _ {l}\right) _ {n, n + 1} \left(\tilde {\mathbf {P}} _ {l} ^ {- 1}\right) _ {n + 1, n + 1} = w _ {j} ^ {- n + 1} w _ {j} ^ {n} = w _ {j}  , & m = n + 1  . \end{array} \right.
$$
The thesis follows from assembling the direct sums back.
Lemma E.2 leads to the following result that fully characterizes the solution of (14) in terms of the generalized eigenvectors and eigenvalues of  $\mathbf{M}$  and  $\mathbf{W}$ .
Proposition E.3. Consider (14) with  $\mathbf{M} = \mathbf{P}\mathbf{J}\mathbf{P}^{-1}$  and  $\mathbf{W}\otimes \mathbf{J} = \tilde{\mathbf{P}}\tilde{\mathbf{J}}\tilde{\mathbf{P}}^{-1}$ , where  $\tilde{\mathbf{J}}$  and  $\tilde{\mathbf{P}}$  are given in Lemma E.2. The solution of (14) is
$$
\operatorname {v e c} (\mathbf {x}) (t) = \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 1} ^ {m} \exp \left(\lambda_ {l _ {1}} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {M}) t\right) \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j}}{(i - j) !} \left(\lambda_ {l _ {1}} (\mathbf {W})\right) ^ {1 - j} \mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {M}),
$$
where the coefficients  $c_{l_1,l_2}^i$  are given by
$$
\operatorname {v e c} (\mathbf {x} _ {0}) = \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 1} ^ {m} \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} (\mathbf {I} \otimes \mathbf {P}) \tilde {\mathbf {P}} \mathbf {e} _ {l _ {1}} \otimes \mathbf {e} _ {l _ {2}} ^ {i}
$$
where  $\{\mathbf{e}_{l_2}^i : l_2 \in \{1, \ldots, m\}, i \in \{1, \ldots, k_{l_2}\}\}$  is the standard basis satisfying  $\mathbf{Pe}_{l_2}^i = \psi_{l_2}^i(\mathbf{M})$ .
Proof. By Lemma E.2, the eigenvalues of  $\mathbf{W} \otimes \mathbf{M}$  and the corresponding eigenvectors and generalized eigenvectors are