# Does nuclear norm decrease as some elements in a matrix are set zero?

Let $$M$$ be a generic nonzero real matrix and $$M_{0}$$ is constructed by replacing some elements in $$M$$ with zero. Let $$||\cdot||_{*}$$ denotes the nuclear norm operator. Is it true that $$||M||_{*}\geq ||M_{0}||_{*}$$ regardless of the replacement rule?

No. Consider $$A=\pmatrix{1&1&1\\ 1&1&1\\ 1&1&1},\ B=\pmatrix{1&0&1\\ 0&1&1\\ 1&1&1}.$$ Since $$A$$ is positive semidefinite, it nuclear norm is equal to its trace, which is $$3$$. The matrix $$B$$ is symmetric but indefinite. Hence its nuclear norm is the sum of absolute values of all its eigenvalues, i.e. $$|1|+|1+\sqrt{2}|+|1-\sqrt{2}|=1+2\sqrt{2}>3$$.
P.S. I asked a similar question before about the induced $$2$$-norm (and the answer is also negative).