# On maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices

## TL;DR

This paper investigates maximum volume submatrices in symmetric semidefinite and diagonally dominant matrices, providing bounds on approximation errors and enhancing cross approximation methods.

## Contribution

It proves that maximum volume submatrices can be principal in certain matrices and extends error bounds for cross approximation methods.

## Key findings

- Maximum volume submatrices can be principal in symmetric semidefinite and diagonally dominant matrices.
- Error bounds for cross approximation are extended to more general matrices.
- For doubly diagonally dominant matrices, the approximation error is within a modest factor of the best.

## Abstract

The problem of finding a $k \times k$ submatrix of maximum volume of a matrix $A$ is of interest in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of $A$. We show that such a submatrix can always be chosen to be a principal submatrix if $A$ is symmetric semidefinite or diagonally dominant. Then we analyze the low-rank approximation error returned by a greedy method for volume maximization, cross approximation with complete pivoting. Our bound for general matrices extends an existing result for symmetric semidefinite matrices and yields new error estimates for diagonally dominant matrices. In particular, for doubly diagonally dominant matrices the error is shown to remain within a modest factor of the best approximation error. We also illustrate how the application of our results to cross approximation for functions leads to new and better convergence results.