What happens to the norm of a matrix \(A\) when some of its elements are replaced by zeros? Here this question is considered for unitarily invariant norms and for such operations on \(A\) as triangular truncation, “pinching” (i.e., \(\sum^k_{j=1}P_jAP_j\), where \(P_j\) are orthogonal projectors in \({\mathbb C}^n\) with orthogonal ranges and \(P_1+\dots+P_k=I\)), generalized diagonalization or tridiagonalization, and trimming (i.e., replacement of all elements of \(A\) outside the band \(-k\leqslant j\leqslant k\) by zeros). Several inequalities for \(A\) and its averages are obtained, and some of them are proved to be sharp.