The Gauss-Seidel method (called Seidel's method by Jeffreys and Jeffreys 1988, p. 305) is a technique for solving the equations of the linear
system of equations one at a time in sequence, and uses previously computed
results as soon as they are available,

There are two important characteristics of the Gauss-Seidel method should be noted. Firstly, the computations appear to be serial. Since each component of the new iterate
depends upon all previously computed components, the updates cannot be done simultaneously
as in the Jacobi method. Secondly, the new iterate
depends upon the order in which the equations are examined. If this ordering is changed,
the components of the new iterates (and not just their order) will also change.

In terms of matrices, the definition of the Gauss-Seidel method can be expressed as