Is it true that Jacobi's method converges for all M- matrices? Definition: $A$ is an M-matrix IFF there exists $B\geq 0$(has the same dimension as A and nonnegative) and $r\in \mathbb{R}$ and $r\geq \rho(B)$($\rho$ is the spectral radius), such that $A=rI-B$ and $A^{-1}\geq 0$.
The iterative method of the system $Ax=b$ converges for any starting vector if $\rho(M^{-1}N)<1$, where $A=M-N$.
In Jacobi's method, if $A=L+D+U$ where $D$ is the diagonal matrix; $L$ and $U$ are  strictly lower triangular matrix and upper triangular matrix, respectively, then $M=D$ and $N=-(L+U)$. 
How to show $\rho(M^{-1}N)<1$ for such $M$ and $N$, and $A=rI-B$?
 A: There is a lot of convergence theory of simple iterations for M-matrices. The book by Varga is a good source of results on the subject, which is a nice application of the Perron-Frobenius theory of nonnegative matrices.
A splitting $A=M-N$ (defining the iteration by $Mx_{k+1}=Nx_k+b$) is called regular if $M^{-1}\geq 0$ and $N\geq 0$. You can show that if $A^{-1}\geq 0$ and $A=M-N$ is a regular splitting, then $\rho(M^{-1}N)<1$. This in particular implies that every regular splitting of an M-matrix is convergent because $A$ is an M-matrix if and only if $A-D\leq 0$ and $A^{-1}\geq 0$. Curiously, convergence of every regular splitting is one of the bunch of equivalent conditions for a Z-matrix to be an M-matrix.
To prove the convergence, note that $M^{-1}N=(I+B)^{-1}B$, where $B:=A^{-1}N\geq 0$. Hence the eigenvalues of $M^{-1}N$ are given by $\alpha/(\alpha+1)$, where $\alpha$ are the eigenvalues of $B$. Since the maximal eigenvalue of $B$ is nonnegative and is equal to $\rho(B)$ (by the Perron-Frobenius theorem), the maximal eigenvalue of $M^{-1}N$ is given by $\rho(B)/[\rho(B)+1]<1$, that is, $\rho(M^{-1}N)<1$ and hence the splitting is convergent (note that the negative eigenvalues of $B$ do not need to bother us since they are bounded from above by $\rho(B)$ in absolute value).
It is easy to see that the Jacobi iteration gives a regular splitting, and hence is convergent for any M-matrix.
