Suppose $\{Q_{nj}\in\mathbb{R}^{p\times p}\}_{n, j\in\mathbb{N}\cup\{0\}}$ are invertible, $Q_{nj}^{\sf{T}} = Q_{jn}$ and every $Q_{nn}$ is positive-definite and $\{s_n\in\mathbb{R}^p\}_{n\in\mathbb{N\cup\{0\}}}$ are bounded (with respect to some norm, say, the $\ell^2$-norm). I wish to find the solution (and its properties) to an infinite system of equations with respect to the infinite unknowns $\{u_{n}\in\mathbb{R}^{p}\}_{n\in\mathbb{N}\cup\{0\}}$ given by the infinite block matrix $$ \begin{bmatrix} Q_{00} & Q_{01} & Q_{02} & \cdots\\ Q_{10} & Q_{11} & Q_{12} & \cdots\\ Q_{20} & Q_{21} & Q_{22} & \cdots\\ \vdots & \vdots & \vdots & \ddots \end{bmatrix} \begin{bmatrix} u_0\\ u_1\\ u_2\\ \vdots \end{bmatrix} = \begin{bmatrix} s_0\\ s_1\\ s_2\\ \vdots \end{bmatrix}. $$ Note: $Q_{nj}$ does not refer to the entry at $(n, j)$ of some matrix $Q$ but rather the $(n, j)$-term of an infinite sequence of $p\times p$ matrices.
The goal is to prove the existence and uniqueness of the solution and the conditions under which the solution is bounded.
My Thought Process: It is obvious that the block matrix is symmetric. Furthermore, I know that $Q_{nn}$ is positive-definite for all $n\in\mathbb{N}\cup\{0\}$. If I truncate the system upto $(n + 1)$ terms, i.e., the following: $$ \begin{bmatrix} Q_{00} & Q_{01} & Q_{02} & \cdots & Q_{0n}\\ Q_{10} & Q_{11} & Q_{12} & \cdots & Q_{1n}\\ Q_{20} & Q_{21} & Q_{22} & \cdots & Q_{2n}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ Q_{n0} & Q_{n1} & Q_{n2} & \cdots & Q_{nn}\\ \end{bmatrix} \begin{bmatrix} u_0\\ u_1\\ u_2\\ \vdots\\ u_n \end{bmatrix} = \begin{bmatrix} s_0\\ s_1\\ s_2\\ \vdots\\ s_n \end{bmatrix} $$ and call the truncated block matrix $Q^{(n)}$ and the corresponding solution $u^{(n)}$, it should be possible to prove by induction that every $Q^{(n)}$ is invertible since every diagonal block is invertible (as it is a positive-definite symmetric matrix) and I know by the forms of $Q_{nj}$ (how I have defined them) that the Schur complement of $Q_{nn}$ in $Q^{(n)}$ is invertible. So in this process, I will have a sequence of solutions $\{u^{(n)}\}_{n = 0}^{\infty}$. Now the goal is to prove that this sequence indeed converges and find the conditions under which the solution is bounded. The boundedness of the limit should be easy to prove if convergence and boundedness of the sequence of solutions is established.
Questions:
- How can I prove that the sequence $u^{(n)}$ converges?
- What conditions may be required for the limit to be bounded?
I do not know at this point if any other property of the matrices $Q_{nj}$ are relevant. These matrices are not arbitrary and have a particular form. Let me know if any other information is required and I will post the definitions of $Q_{nj}$.