Divide-and-conquer Eigenvalue Algorithm - Divide

The divide part of the divide-and-conquer algorithm comes from the realization that a tridiagonal matrix is "almost" block diagonal.

The size of submatrix we will call, and then is . Note that the remark about being almost block diagonal is true regardless of how is chosen (i.e., there are many ways to so decompose the matrix). However, it makes sense, from an efficiency standpoint, to choose .

We write as a block diagonal matrix, plus a rank-1 correction:

The only difference between and is that the lower right entry in has been replaced with and similarly, in the top left entry has been replaced with .

The remainder of the divide step is to solve for the eigenvalues (and if desired the eigenvectors) of and, that is to find the diagonalizations and . This can be accomplished with recursive calls to the divide-and-conquer algorithm, although practical implementations often switch to the QR algorithm for small enough submatrices.