On the additive structure of the inverses of banded matrices
Using the results of \textit{W. W. Barrett} and \textit{P. J. Feinsilver} [ibid. 41, 111-130 (1981; Zbl 0472.15004)] the author proves the following main result: Let B be an \(n\times n\) proper \((2k+1)\)-diagonal matrix. Then \(B^{-1}=\sum^{k}_{p=1}[(u_ pv^ T_ p)\circ E_ n+(x_ py^ T_ p)\circ L_ n],\) where \(u_ p,v_ p,x_ p,y_ p\) are n-vectors, \(E_ n\underline\triangle (e_{ij})\), \(e_{ij}\underline\triangle 1\) if \(i\leq j\), and \(\underline\triangle 0\) if \(i>j\), \(L_ n\underline\triangle E^ T_ n-I_ n\) \((I_ n\) is the unit matrix) and \(\circ\) denotes the Hadamard product. If in addition B is symmetric, then \(B^{-1}\) may be expressed as a sum of k symmetric matrices belonging to the class of inverses of symmetric irreducible tridiagonal matrices. Two examples are given. The results may be applied for the resolution of constant coefficient banded linear systems in VLSI models.
symmetric irreducible tridiagonal matrices
scientific article; zbMATH DE number 3964052