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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 aquasi-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 submatrixif $A$ is symmetric semidefinite or diagonally dominant. Then we analyze thelow-rank approximation error returned by a greedy method for volumemaximization, cross approximation with complete pivoting. Our bound for generalmatrices extends an existing result for symmetric semidefinite matrices andyields new error estimates for diagonally dominant matrices. In particular, fordoubly diagonally dominant matrices the error is shown to remain within amodest factor of the best approximation error. We also illustrate how theapplication of our results to cross approximation for functions leads to newand better convergence results.

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