The matrix of B with respect to any limit point of this sequence is diagonal. Of course no classical mathematician would state the principal axes in the second form, but this form contains all the general computational content of the theorem. One can say more if the characteristic polynomial of B , which is the same in any coordinate system, has simple roots. Each simple root determines a principal axis. The problem comes from the possibility of multiple roots.
B can cause large changes in the principal axes.
The worst case is when a matrix for B looks like an identity matrix, because that gives no information about what the principal axes might be. Note that there is no problem writing down the desired matrix because the diagonal elements are the roots of the characteristic polynomial, and these can be computed. The problem is finding the coordinate system that diagonalizes the matrix. In an interesting reinterpretation of what that constructive proof shows, he establishes a classical result regarding diagonalizing matrices over rings of continuous functions on certain kinds of topological spaces. The principal axes theorem is related to the problem of finding a point in a compact set at which a given uniformly continuous function achieves its maximum.