On the minimum trace norm of (0,1)-matrices

The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm $psi_{n}left( mright) $ of $left( 0,1right) $-matrices of size $ntimes n$ with exactly $m$ ones. It is shown that: (1) if $ngeq2$ and $n<mleq2n,$ then $psi_{n}left( mright) leq sqrt{m+sqrt{2left( m-1right) }}$ , with equality if and only if $m$ is a prime; (2) if $ngeq4$ and $2n<mleq3n,$ then $psi_{n}left( mright) leq sqrt{m+2sqrt{2leftlfloor m/3rightrfloor }}$ , with equality if and only if $m$ is a prime or a double of a prime; (3) if $3n<mleq4n,$ then $psi_{n}left( mright) leqsqrt{m+2sqrt{m-2}}% $ , with equality if and only if there is an integer $kgeq1$ such that $m=12kpm2$ and $4kpm1,6kpm1,12kpm1$ are primes.