## 4 Convergence and Rates
Now write  
\[= \frac{1}{J}\sum_{j=1}^{J}(X_{ij})^{2}-\frac{2}{J}\sum_{j=1}^{J}X_ {ij}\bar{X}_{i\cdot}^{(0)}+(\bar{X}_{i\cdot}^{(0)})^{2}\]4 \[= \frac{1}{J}\sum_{j=1}^{J}(X_{ij})^{2}-\frac{2}{J}\sum_{j=1}^{J}X_ {ij}\bar{X}_{i\cdot}^{(0)}+(\bar{X}_{i\cdot}^{(0)})^{2}\]5 \[= \frac{1}{J}\sum_{j=1}^{J}(X_{ij})^{2}-\frac{2}{J}\sum_{j=1}^{J}X_ {ij}\bar{X}_{i\cdot}^{(0)}+(\bar{X}_{i\cdot}^{(0)})^{2}\]6 \[= \frac{1}{J}\sum_{j=1}^{J}(X_{ij})^{2}-\frac{2}{J}\sum_{j=1}^{J}X_ {ij}\bar{X}_{i\cdot}^{(0)}+(\bar{X}_{i\cdot}^{(0)})^{2}\]7 \[= \frac{1}{J}\sum_{j=1}^{J}(X_{ij})^{2}-\frac{2}{J}\sum_{j=1}^{J}X_ {ij}\bar{X}_{i\cdot}^{(0)}+(\bar{X}_{i\cdot}^{(0)})^{2}\]8  
One argues that  
\[= \frac{1}{J}\sum_{j=1}^{J}(X_{ij})^{2}-\frac{2}{J}\sum_{j=1}^{J}X_ {ij}\bar{X}_{i\cdot}^{(0)}+(\bar{X}_{i\cdot}^{(0)})^{2}\]9where \(D=D(I,J)<\infty\). A key observation is that for every \(m\), there exists \(j=j(m)\) for which \(\{\lambda_{ij}^{(2m-1)}\}\) are not of the same or all of the opposite sign as \(\{Z_{ij}:i=1,...,I\}\), which, as was noted, are not, themselves of the same sign. Argue analogously regarding \(\{\lambda_{ij}^{(2m)}\}\). Refer now to \[= \frac{1}{J}\sum_{j=1}^{J}(X_{ij})^{2}-2\bar{X}_{\cdot j}\bar{X}_ {i\cdot}+(\bar{X}_{i\cdot})^{2}. \tag{3}\]0 Our arguments show the correctness of the concentric circles in \(\mathbb{R}^{IJ}\) on which \(\mathbf{X}^{(n)},\mathbf{X}^{(n+1)},\mathbf{X}^{(n+2)},\cdots\) ultimately lie. We conclude that to suitable approximation, the successive iterates lie on such circles with radii in geometric ratio. That ratio is uniformly \(<1\), but is _not_ arbitrarily close to \(0\). However Figure 9 buries a key idea in our study of convergence are rates. Thus, write \(\theta^{(n)}\) for the angle between \(X^{(n)}\) and \(Z\). Obviously, the squared Frobenius norm \(\|\mathbf{X}^{(n)}-\mathbf{Z}\|_{F}^{2}\) can be expressed as  
\[= \frac{1}{J}\sum_{j=1}^{J}(X_{ij})^{2}-2\bar{X}_{\cdot j}\bar{X}_ {i\cdot}+(\bar{X}_{i\cdot})^{2}. \tag{3}\]1  
Now rewrite \(\cos(\theta^{(n)})=1-\frac{1}{2 IJ}(\|\mathbf{X}^{(n)}-\mathbf{Z}\|_{F}^{2})\). Therefore, convergence of \(\mathbf{X}^{(n)}\) to \(\mathbf{Z}\) implies that \(\theta^{(n)}\) can be taken arbitrarily close to \(0\). This is another way of saying that for each \(n\), \(\|\mathbf{X}^{(n)}\|_{F}^{2}=IJ\). Failure of this condition is why \"simultaneous normalization\" fails. **\[= \frac{1}{J}\sum_{j=1}^{J}(X_{ij})^{2}-2\bar{X}_{\cdot j}\bar{X}_ {i\cdot}+(\bar{X}_{i\cdot})^{2}. \tag{3}\]2** For \(n\) large enough \(\log\{\|\mathbf{X}^{(n+2)}-\mathbf{X}^{(n+1)}\|_{F}^{2}/\|\mathbf{X}^{(n+1)}- \mathbf{X}^{(n)}\|_{F}^{2}\}<1\) uniformly. **5. Illustrations of Convergence**  
We now illustrate the rapidity of convergence in the \(3\times 3\) case to give a geometric perspective. First note that in the \(3\times 3\) case the set of fixed points are characterized by 3 unique values. For instance, the following doubly normalized matrix which arises a result of successive normalization has only three unique elements. \[\mathbf{X}^{(\textit{final})} = \begin{bmatrix}-1.4137&0.7407&0.6730\\ 0.7407&0.6730&-1.4137\\ 0.6730&-1.4137&0.7407\end{bmatrix}  \tag{4}\]  
Hence we can use the first column of the limit matrix to represent the fixed point arising from successive normalization. Hence the curve of fixed points can be generated by applying the successive