Matrix continued fractions

Question

I am aware of the classical continued fraction in the field of real numbers, but recently I have come across the term matrix continued fraction and when I checked on the internet there are varieties of definition for the same and I am interested in exploring this direction further for my research.

I have the following questions.

1. What are some papers where I can start with?

2. What are some branches of Mathematics where these continued fractions are applied?

Thanks for your time. Have a good day.

Answer 1

The following is essentially a contribution to point 2. Non-commutative (which can be specialized to matrices or scalars) continued fraction are used in enumeration and language theories. Two examples : Dyck and Motzkin paths.

These are two lattice paths drawn on $\mathbb{N}^2$ (the first quarter-plane).

Dyck Paths Steps are $a=(1,1)$ (north-east step) and $b=(1,-1)$ (south-east step) Dyck paths are defined as paths that

1. start at $(0,0)$
2. end at $(0,2n)$ (for some $n$)
3. always stand above the $x$-axis

They can be coded by words $w$ (Dyck words) in the alphabet $\{a,b\}$ such that, if $w=uv$ one has $|u|_a\geq |u|_b$ (the path is always in $\mathbb{N}^2$) and, at the end $|w|_a=|w|_b$ (it returns to the $x$-axis). In what preceeds $|u|_a$ (resp. $|u|_b$) stand for the number of occurences of $a$ (resp. $b$) in the word $u$.

Let $D$ be the set of Dyck words. It can be shown that it is a free monoid with alphabet (irreducible Dyck) $A$ (the paths which return to the $x$-axis only at the end). Indentifying them with their characteristic series $$ D=\sum_{w\atop\small{Dyck\ word}}\,w\ ; A=\sum_{w\atop\small{Irreducible\ Dyck\ word}}\,w $$ (computed in $\mathbb{Z}\langle\langle a,b\rangle\rangle$) one has $A=aDb$ and $D=(1-A)^{-1}$, then
$$ A=a(1-A)^{-1}b=a\frac{1}{1-A}b=a\frac{1}{1-a\frac{1}{1-A}b}b=a\frac{1}{1-a\frac{1}{1-a\frac{1}{1-A}b}b}b=\ldots $$ with suitable structures, one can show that this continued fraction converges.

Motzkin Paths Similarly, but with steps are $a=(1,1)$ (north-east step), $c=(1,0)$ (east step, i.e. constant) $b=(1,-1)$ (south-east step) Motzkin paths are defined as paths that

1. start at $(0,0)$
2. end at $(0,n)$ (for some $n$)
3. always stand above the $x$-axis

They are coded by Motzkin words which form again a free monoid $M$ with alphabet, say $B$ of (irreducible Motzkin words). One has $B=(c+aMb)$ and $M=(1-B)^{-1}$ thus $$ B=c+aMb=c+a\frac{1}{(1-B)}b=\ldots $$ Hope this helps. Do not hesitate to interact.