Is it possible to remove $k$ zero eigenvalues of a matrix by removing $k$ rows and columns of a square matrix (after modification using decomposition methods or other ways) with other eigenvalues unchanged?. If so, how to identify the rows and column to be removed?
Major problem is the loss of sparsity on modified matrix
PS: A simple understanding is, with one zero eigenvalue, there is one dependent row and column in the matrix. Then the dependent row/column can be written as a linear combination of other rows/columns and can be used to reduce the dimension using a transformation. But not sure the eigenvalues invariability and how to choose the the dependent row/column (there will be multiple possibility).
Best Answer
No. Let $A = \left[\begin{array}{cc}2 & 2\\2 & 2\end{array}\right]$. This matrix has eigenvalues $0$ and $4$.