# 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.

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

Thanks for your time. Have a good day.

• For the first question, Hamilton considered continued fractions with quaternion coefficients (which may be viewed as structured $4\times 4$ matrices over the reals) as early as 1852 (see maths.tcd.ie/pub/HistMath/People/Hamilton/ConFrac/ConFrac.pdf). Lorentzen recently gave a talk which announces results on more general continued fractions here (pg 81): cmft2017.umcs.lublin.pl/wp-content/uploads/2017/07/cmft2017.pdf Lorentzen and Waadeland's book "Continued Fractions" explains some applications to differential equations and orthogonal polynomials for example. – Josiah Park Mar 19 '18 at 20:03
• Some references on multidimensional continued fractions: Arnold, V. I., Regul. Chaotic Dyn. 3 (1998), no. 3, 10–17. Arnold, V. I., Comm. Pure Appl. Math. 42 (1989), no. 7, 993–1000. Kontsevich, M. L.; Suhov, Yu. M., Pseudoperiodic topology, 9–27, Amer. Math. Soc. Transl. Ser. 2, 197, Adv. Math. Sci., 46, Amer. Math. Soc., Providence, RI, 1999. Karpenkov, O. N. Math. Comp. 78 (2009), no. 267, 1687–1711. Lachaud, G., C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 8, 711–716. Korkina, E., C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 8, 777–780, – Dan Fox Mar 21 '18 at 7:47

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.