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.

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 \( \mathbf{Q} \) and an upper triangular matrix \( \mathbf{U} \). Historically R was used instead of U since the wording right triangular matrix was first used. The method is based on an iterative procedure similar to Jacobi's method, by a succession of planar rotations. For a tridiagonal matrix it is simple to carry out in principle, but complicated in detail! We will discuss this in more detail during week 38.