Decay Rates of the Inverse of Nonsymmetric Tridiagonal and Band Matrices

It is well known that the inverse C = c i;j ] of an irreducible nonsingular symmetric tridiagonal matrix is a Green matrix, i.e. it satisses c i;j = u i v j for i j and two sequences of real numbers fu i g and fv i g. A similar result holds for nonsymmetric matrices A. There the inverse is given by four sequences fu i g; fv i g; fy i g; and fy i g: Here we characterize certain properties of A in terms of the u i ; v i ; x i , and y i namely that A is an M{matrix or positive deenite. We also establish a relation of zero row sums and zero column sums of A and pairwise constant u i ; v i ; x i and y i. Moreover we consider decay rates for the entries of the inverse of tridiagonal and block tridiagonal (banded) matrices. For diagonally dominant matrices we show that the entries of the inverse strictly decay along a row or column. We give a sharp decay result for tridiagonal irreducible M{matrices and tridiagonal positive deenite matrices. We also give a decay rate for arbitrary banded M{matrices.