Factorization of banded lower triangular infinite matrices (Q2564966)

An infinite matrix \((t_{jk})\) is called lower triangular strictly \(m\)-banded if \(t_{jk}=0\) for \(k>j\) and for \(k<j-m\), but \(t_{ii}\), \(t_{i,i-m}\neq 0\) for all \(i\). The authors show that every such matrix is a product of \(m\) lower triangular strictly 1-banded matrices. Their proof is based on a representation of the \(m\)-banded matrix \(T\) as an input-output map of a linear finite-dimensional time-varying system.