If we now choose \( \theta \) to give the element \( a_{21}^{'} = e^{'}= 0 \) then we have the first eigenvalue \( = a_{11}^{'} = d_{1}^{'} \).

This procedure can be continued on the remaining three-dimensional submatrix for the next eigenvalue. Thus after few transformations we have the wanted diagonal form.

What we see here is just a special case of the more general procedure developed by Francis in two articles in 1961 and 1962. Using Jacobi's method is not very efficient ether.

The algorithm is based on the so-called QR method (or just QR-algorithm). It follows from a theorem by Schur which states that any square matrix can be written out in terms of an orthogonal matrix \( \hat{Q} \) and an upper triangular matrix \( \hat{U} \). Historically \( R \) was used instead of \( U \) since the wording right triangular matrix was first used.