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Finding the $r\times r$ submatrix of maximum volume of a matrix $A\in\mathbbR^{n\times n}$ is an NP hard problem that arises in a variety of applications.We propose a new greedy algorithm of cost $\mathcal O(n)$, for the case $A$symmetric positive semidefinite (SPSD) and we discuss its extension to relatedoptimization problems such as the maximum ratio of volumes. In the second partof the paper we prove that any SPSD matrix admits a cross approximation builton a principal submatrix whose approximation error is bounded by $(r+1)$ timesthe error of the best rank $r$ approximation in the nuclear norm. In the spiritof recent work by Cortinovis and Kressner we derive some deterministicalgorithms which are capable to retrieve a quasi optimal cross approximationwith cost $\mathcal O(n^3)$.

Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices