# Change in eigenvector when new rows and columns are added to a matrix

I have a matrix $$\mathbf{X}$$ and I use the eigenvectors of the symmetric matrix $$\mathbf{X}^{\top}\mathbf{X}$$ for further analysis. But the $$\mathbf{X}$$ is always growing, as new data points are added to it. I do not want to calculate the eigenvectors each time new rows are added to $$\mathbf{X}$$. One fairly good assumption regarding the new rows are: in most cases the newly added rows are linear combinations of the existing rows. But in rare cases, it can be independent also.

Is there any quantification of the changes in the eigenvectors of $$\mathbf{X^\top X}$$ as new rows are added to $$\mathbf{X}$$ ?. Any one can point me to me related papers ?

• this seems related to the Bunch-Nielsen-Sorensen formula – tch Jan 12 at 19:09
• @tch, in that specific problem, the dimension of the matrix remains same. I think it is equivalent to changing entries of a symmetric matrix. But in my case, the matrix is incrementally builds by adding new rows and columns. – Shew Jan 13 at 12:14
• But the matrix $X^TX$ stays the same size even if X gets new rows. In particular $X^TX=\sum_i x_i x_i^T$ where $x_i$ are the rows of $X$. – tch Jan 13 at 15:38