Bounds Matrix And M Matrix Determined

This paper is concerned in three important matrices classes: tridiagonal matrices, nonnegative matrices and M matrices. Through the studies of some quantities and characteristics about those, some good conclusions are obtained.Firstly, new upper and lower bounds of inverse elements of tridiagonal matrices is presented only assuming that the tridiagonal matrices satisfy the appropriate condition (see the following condition (H)). For strictly diagonally dominant matrices, this bounds are sharper than those of [1, Lin. Alg. Appl., 287(1999), 289-305] and [2, Lin. Alg. Appl., 330 (2001), 1-14].Secondly, a new matrix Ac = A -ρ( A ) uu^T(1 + c) is introduced. Making use of its beautiful properties, the improved lower and upper bounds for the nonmaximal eigenvalue of a symmetric positive matrix are presented. The bounds are sharper than the bounds of Zhang and Luo in [3, Czechoslovak Math. J., 52 (127) (2002): 537-544].Then, using the definition of Schur complement, an effective method for judging nonsingular M matrix is gived. This judging method is equivalent to corresponding method in [4, Numerical Mathematics a Journal of Chinese Universities, 2001, 4: 357-362] and more simple than it.Finally, an estimate for q( A(?)B^-1)from blow that improves Fiedler and Markham's result in [5, Lin. Alg. Appl., 101(1988), 1-8] is derived and easily computable lower and upper bounds of q( A(?)B^-1) for strictly diagonally dominant M matrices A and B is established.