Let M
be a real positive semidefinte matrix and consider the entrywise nonnegative matrix M'
obtained from from M
by zeroing out all the negative entries of M
. Is it true that M'
is always positive semidefinite?
Addendum 1: More generally, consider the entrywise nonnegative matrix M''
obtained from M
by zeroing out an arbitrary set of off-diagonal entries (symmetrically, of course). Is it true that M''
is always positive semidefinite?
Addendum 2: Thanks to @orangeskid and @user1551 for prompt answers. The question of Addendum 1 has a counterexample even in 3 dimensions.