Matrix continued fractions 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.


*

*What are some papers where I can start with?

*What are some branches of Mathematics where these continued fractions are applied? 
Thanks for your time. Have a good day.
 A: 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 


*

* start at $(0,0)$ 

* end at $(0,2n)$ (for some $n$)

* 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 


*

* start at $(0,0)$ 

* end at $(0,n)$ (for some $n$)

* 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.
