A common characterization of
M-matrices are non-singular square
matrices with non-positive
off-diagonal entries, positive
diagonal entries, non-negative row
sums, and at least one positive row
sum
It seems that this characterization depends on the use of rows or columns. I don't understand how that is possible, as the transposed of an M-Matrix is also an M-Matrix.
Example:
$\left( \begin{array}{ccc}
2 & -1 &0\\
-2 & 2&-1\\
0&-1&2 \end{array} \right)$
This matrix is not diagonally dominant, but its transpose is.
However the matrix is clearly M-Matrix, because all Eigenvalues are positive.
Thanks for your help!
Best Answer
There are many characterizations of non-singular M-matrices which make it clear that the property is preserved by taking transposes. For example, Berman and Plemmons book, Nonnegative matrices in the mathematical sciences, devotes Chapt. 6 to M-matrices. Thm. 2.3 there gives many equivalences to a Z-matrix (that is, a matrix with nonpositive off-diagonal entries) to be a non-singular M-matrix. But the condition you've cited is not among them.
One equivalence given there ($A_1$) is for all the principal minors to be positive. Clearly this is preserved by transpose.
The sufficient condition you described is not preserved by taking transpose, as your example shows. However it is certainly an M-matrix. The cited sufficient condition is closely related to weak diagonal dominance, and it seems to be an echo of the result by Minkowski (that a Z-matrix with all positive row sums has positive determinant) which apparently motivated Ostrowski's choice of the term M-matrix (honoring Minkowski).
Added: I've corrected the Wikipedia article. Thanks for pointing out this mistake.