# M-Matrix characterization for the transpose

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.

• Isn't the second row sum negative for your example? – hardmath Jan 12 '11 at 19:02
• yes. its not diagonally dominant. the row sum is negative. and its still a M-Matrix. – Johannes Gerer Jan 12 '11 at 19:30
• I'm afraid Wikipedia has let you down here. Unless one assumes "characterization" was meant to imply a sufficient condition rather that a definition, the statement quoted is not correct. Some authors include singular as well as non-singular M-matrices, but if we restrict attention to the non-singular one, then the first part of the Wikipedia article is correct. – hardmath Jan 12 '11 at 19:41

One equivalence given there ($A_1$) is for all the principal minors to be positive. Clearly this is preserved by transpose.