The Wikipedia article on the Jacobi method states that
This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization.
However, I fail to see why this is the case. Could someone please explain the connection between the Jacobi method and the Jacobi eigenvalue algorithm?
I see no obvious connection between the two methods.
Jacobi's method for solving linear systems is a functional iteration based on a splitting of the matrix $A$ and it converges linearly at best. Each iteration consists of a linear update and a solve. The Jacobi iteration for computing the eigendecomposition of a symmetric matrix $A$ is based on orthogonal similarity transformations and converges quadratically. Each iteration consists of a complete sweep of the entries nullifying off-diagonal pairs as we go along.
Both algorithms suffer from low arithmetic intensity in the sense that they do only a few flops per memory operations. Jacobi's method for linear system has limited potential for blocking, i.e, dense matrix matrix multiplication and is inherently serial. Jacobi's iteration for computing an eigendecomposition has great potential for blocking and parallelization.
If there is a connection between the two methods, then it deserves more than a single line of text. The issue would have been raised during a formal peer review.