# Sufficient conditions for invertibility of a block tridiagonal matrix

Let $$M in mathbb{R}^{N times N}$$ be a block-tridiagonal matrix:

$$M = begin{bmatrix} B_1 & C_1 & 0 & 0 & cdots & 0 \ A_2 & B_2 & C_2 & 0 & cdots & 0 \ 0 & A_3 & B_3 & C_3 & cdots & 0 \ 0 & 0 & A_4 & B_4 & cdots & 0 \ vdots & vdots & vdots & vdots & ddots & vdots \ 0 & 0 & cdots & 0 & A_n & B_n end{bmatrix}$$

where each $$B_i$$ is square and invertible, with varying sizes.

## Problem

What are sufficient conditions on $$A_i$$, $$B_i$$ and $$C_i$$ for showing that $$M$$ is invertible?

For example, a sufficient, but weak condition is that $$C_i = 0$$ for each $$i$$.

### Reformulation by Schur complements

Let

$$D_i = B_i – C_i D_{i + 1}^{-1} A_{i + 1},$$
$$D_n = B_n.$$

Supposing each $$D_i$$ is invertible, by elimination $$M$$ reduces to

$$M sim begin{bmatrix} D_1 & 0 & 0 & 0 & cdots & 0 \ A_2 & D_2 & 0 & 0 & cdots & 0 \ 0 & A_3 & D_3 & 0& cdots & 0 \ 0 & 0 & A_4 & D_4 & cdots & 0 \ vdots & vdots & vdots & vdots & ddots & vdots \ 0 & 0 & cdots & 0 & A_n & D_n end{bmatrix}$$

Then

$$det(M) = prod_{i = 1}^n det(D_i).$$

Hence invertibility of each $$D_i$$ is invertible implies $$M$$ is invertible. The converse also holds.

Hence sufficient conditions for the invertibility of Schur complements are also useful.

### Reformulation by 2×2 block matrices

Let

$$E_i = begin{bmatrix} B_i & C_i \ A_i & D_{i + 1} end{bmatrix},$$
$$E_n = B_n$$

Then

$$det(E_i) = det(D_{i + 1}) det(D_i) = prod_{j = i}^n det(D_i),$$
$$det(E_1) = det(M).$$

Hence sufficient conditions for a 2×2 block matrix with square invertible diagonal blocks being invertible are also useful.