Questions tagged [infinite-matrices]

I have an equation $AX=B$, where $A$ is $\infty \times \infty$ matrix, $X$ is $\infty \times 1$ vector and $B$ is $\infty \times 1$ vector. $A$ and $B$ are known and I need to determine $X$. For this, I think that I should calculate inverse of $A$…

I have heard of infinite-dimensional matrices that have a countably-infinite number of dimension. Is it possible that there could be a matrix with aleph-1, aleph-2, or even aleph-aleph-0 dimensions?

Claim: For any singly infinite non-invertible matrix $A$, let $A$ to be injective and $A=BC$, where $B$ is invertible, and $C$ is a product of shifting matrices. Is this claim true? Any reference would be nice. By shifting matrices, I mean ones…