Let's consider the system $$ \begin{cases} 3x_{1}-x_{2}+x_{3} & =5,\\ 2x_{1}+5x_{2}-2x_{3} & =1,\\ 2x_{1}-x_{2}+6x_{3} & =7. \end{cases} $$ The Gauss-Seidel method is an iterative process: $$ X_{k}=TX_{k-1}+C, $$ where $$ T=\left[\begin{array}{ccc} 3 & 0 & 0\\ 2 & 5 & 0\\ 2 & -1 & 6 \end{array}\right]^{-1}\left[\begin{array}{ccc} 0 & 1 & -1\\ 0 & 0 & 2\\ 0 & 0 & 0 \end{array}\right]. $$ My question: do we have a way to find $\left\Vert T\right\Vert _{\infty}$ without finding $T$ explicitly? I ask this question since in practice by calculating the error by hand, we need to find the norm $\left\Vert T\right\Vert _{\infty}$. It's quite time-consuming to first find $T$, then evaluate $\left\Vert T\right\Vert _{\infty}$. If we have a formula for $\left\Vert T\right\Vert _{\infty}$ associated with the coefficient matrix, it will save time.
Thanks.