  In this document, some elements of the theory and algorithmics corresponding
to the existence and computability of approximate joint eigenpairs for finite
collections of matrices with applications to model order reduction, are
presented. More specifically, given a finite collection $X_1,\ldots,X_d$ of
Hermitian matrices in $\mathbb{C}^{n\times n}$, a positive integer $r\ll n$,
and a collection of complex numbers $\hat{x}_{j,k}\in \mathbb{C}$ for $1\leq
j\leq d$, $1\leq k\leq r$. First, we study the computability of a set of $r$
vectors $w_1,\ldots,w_r\in \mathbb{C}^{n}$, such that $w_k=\arg\min_{w\in
\mathbb{C}^n}\sum_{j=1}^d\|X_jw-\hat{x}_{j,k} w\|^2$ for each $1\leq k \leq r$,
then we present a model order reduction procedure based on the truncated joint
approximate eigenbases computed with the aforementioned techniques. Some
prototypical algorithms together with some numerical examples are presented as
well.
