Inverses of quasi-tridiagonal matrices
A blockmatrix \(T=[T_{ij}]\) with square blocks \(T_{ii}\) is quasi- triangular, if \(T_{ij}=0\) for \(| i-j|>1\). Explicit formulas for the inverses of quasi-triangular matrices are given. The inverses can be characterized by the quasi-triangle property: \(R=[R_{ij}]\) has the q.t.p. if all \(R_{ii}\) nonsingular and \(R_{ij}=R_{ik}R^{- 1}_{kk}R_{kj}\) for all \(i<k<j\) and for all \(i>k>j\). Theorem 3: Let K be a nonsingular block matrix whose diagonal blocks \(K_{ii} (i=2,...,n- 1)\) are nonsingular. Then K has the q.t.p. iff its inverse is quasi- tridiagonal.
scientific article; zbMATH DE number 3853273
scientific article