# Generalized ultrametric matrices — a class of inverse M-matrices

@article{Nabben1995GeneralizedUM, title={Generalized ultrametric matrices — a class of inverse M-matrices}, author={Reinhard Nabben and Richard S. Varga}, journal={Linear Algebra and its Applications}, year={1995}, volume={220}, pages={365-390} }

## 40 Citations

Special Ultrametric Matrices and Graphs

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2000

The rank of a special ultrametric matrix is recognized and it is shown that its Moore--Penrose inverse is a generalized diagonally dominant M-matrix.

Inverse Tridiagonal Z{matrices

- Mathematics

In this paper, we consider matrices whose inverses are tridiagonal Z{matrices. Based on a characterization of symmetric tridiagonal matrices by Gantmacher and Krein, we show that a matrix is the…

Matrices whose inverses are generalized M-matrices

- Mathematics
- 1998

In 1972 T. L. Markham defined the so-called matrices of type-D [M]. Markham proved that the inverse of a certain type-D matrix is a tridiagonal M-matrix. Recently, several generalizations of this…

Inverse tridiagonal Z-matrices

- Mathematics
- 1998

In this paper, we consider matrices whose inverses are tridiagonal Z--matrices. Based on a characterization of symmetric tridiagonal matrices by Gantmacher and Krein, we show that a matrix is the…

A polynomial time spectral decomposition test for certain classes of inverse M-matrices

- Mathematics
- 1998

The primary result in this paper is a set of O(n) time algorithms to determine whether a speci ed real, symmetric matrix is a member of any of several closely related classes: the MMA-matrices; the…

Hadamard Functions of Inverse M-Matrices

- MathematicsSIAM J. Matrix Anal. Appl.
- 2009

It is proved that the class of generalized ultrametric matrices (GUM), which includes naturally the GUM matrices, is the largest class of bipotential matrices stable under Hadamard increasing functions.

Inverse tridiagonal Z-Martices ∗

- Mathematics
- 1998

In this paper, we consider whose invereses are tridiagonal Z-matrices Based on a characterization of symmetric tridiagonal matirices by Gantmacher and Kein, we show that a matrix is the inverse of a…

On Green's Matrices of Trees

- MathematicsSIAM J. Matrix Anal. Appl.
- 2001

It is proved that the inverses of irreducible acyclic symmetric matrices are given as the Hadamard product of three matrices, a type D matrix, a flipped type D Matrix, and a matrix of tree structure which is closely related to the graph of the original matrix itself.

## References

SHOWING 1-10 OF 11 REFERENCES

A Linear Algebra Proof that the Inverse of a Strictly UltrametricMatrix is a Strictly Diagonally Dominant Stieltjes Matrix

- Mathematics
- 1994

It is well known that every $n \times n$ Stieltjes matrix has an inverse that is an $n \times n$ nonsingular symmetric matrix with nonnegative entries, and it is also easily seen that the converse of…

On the inverse M-matrix problem for real symmetric positive-definite Toeplitz matrices

- Mathematics
- 1991

Necessary and sufficient conditions are obtained for a real symmetric positive-definite Toeplitz matrix R to be an inverse of an M-matrix in terms of its Schur coefficients. Related problems are also…

Inverse of Strictly Ultrametric Matrices are of Stieltjes Type

- Mathematics
- 1994

This paper shows that a nonnegative ultrametric matrix $A$ is nonsingular and that its inverse is a strictly diagonally dominant Stieltjes matrix. The method consists of studying the spectral…

Nonnegative matrices whose inverses are M-matrices

- Mathematics
- 1972

A characterization of a class of totally nonnegative matrices whose inverses are M-matrices is given. It is then shown that if A is nonnegative of order n and A-1 is an M-matrix, then the almost…