Bounds for norms of the matrix inverse and the smallest singular value Original Research Article

In the first part, we obtain two easily calculable lower bounds for double vertical barA-1double vertical bar, where double vertical bar·double vertical bar is an arbitrary matrix norm, in the case when A is an M-matrix, using first row sums and then column sums. Using those results, we obtain the characterization of M-matrices whose inverses are stochastic matrices. With different approach, we give another easily calculable lower bounds for double vertical barA-1double vertical bar∞ and double vertical barA-1double vertical bar1 in the case when A is an M-matrix. In the second part, using the results from the first part, we obtain our main result, an easily calculable upper bound for double vertical barA-1double vertical bar1 in the case when A is an SDD matrix, thus improving the known bound. All mentioned norm bounds can be used for bounding the smallest singular value of a matrix.