Convergence of Gauss-Seidel for a matrix where the transposed matrix is strictly diagonally dominant In this post, it is shown that for a matrix $A\in\mathbb R^{n\times n}$, the Gauss-Seidel iteration method for the solution of linear systems converges if $A$ is strictly diagonally dominant, i.e. if $$|a_{ii}|>\sum_{j=1\\j\neq i}^n|a_{ij}|$$ for all $i=1,\dots,n$. (One has to show that for $T:=-(D+L)^{-1} U$, given the splitting of $A$ in lower, diagonal, and upper triangular matrices, $A=L+D+U$, the spectral radius satisfies $\rho(T)<1$.)
A similar result is true if $A^\intercal$ is strictly diagonally dominant, i.e. if $$|a_{ii}|>\sum_{j=1\\j\neq i}^n|a_{ji}|$$ for all $i=1,\dots,n$. Is there any easy way to use the first result to derive the second?
 A: Yes, it is possible to apply the result that has already been established.

Let $A = M - N$ be the splitting of $A$ where $M = D + L$ is lower triangular and $N = -U$ is strictly upper triangular and let $G = M^{-1}N$ be the matrix driving the functional iteration 
$$
M x_{n+1} = Nx_n + f
$$
If $A$ is strictly diagonally dominant by rows, then we already know that the spectral radius of $G$ satisfies $\rho(G) < 1$. We will now assume that the matrix $A$ is strictly diagonally dominant by columns. We now define an auxiliary sequence $\{y_n\}$ by $$y_n = M x_n.$$ The sequence $\{x_n\}$ is convergent if and only the sequence $\{y_n\}$ is convergent because $M$ is nonsingular. By definition, $y_n$ satisfies the functional iteration
$$ y_{n+1} = N M^{-1} y_n + f$$
We will know show that this sequence is convergent. Let $H = NM^{-1}$ denote the central matrix. We claim that $\rho(H) < 1$. It is enough to show that $$\rho(H^T) < 1.$$
By assumption, $A^T$ is strictly diagonally dominant by rows and $A^T = M^T - N^T$ is immediate. It follows that the matrix $(M^T)^{-1} N^T$ satisfies $$\rho((M^T)^{-1} N^T) < 1.$$ Howewer $$(M^T)^{-1} N^T = (M^{-1})^T N^T = (N M^{-1})^T = H^T.$$ This shows that $\rho(H) < 1$. It follows that $\{y_n\}$ is convergent and the limit $y$  satisfies $$y = NM^{-1} y + f.$$ It follows that $\{x_n\}$ is convergent with limit $x = M^{-1} y$ that satisfies $$Mx = Nx + f.$$
This completes the analysis.

The Jacobi iteration is discussed in here 
https://math.stackexchange.com/a/2813489/307944
