$A^{\dagger}A\geq B^{\dagger}B$. Can we say that $|Tr[A]|\geq |Tr[B]|$?

Question

Suppose that A and B are any two square matrices of equal dimension with complex entries and $A^{\dagger}A\geq B^{\dagger}B$. Can we say that $|Tr[A]|\geq |Tr[B]|$?

Answer 1

Try $$A = \pmatrix{0 & 1\cr 1 & 0\cr},\ B = \pmatrix{1 & 0\cr 0 & 1\cr}$$ which have $A^\dagger A = B^\dagger B$, but $\text{Tr}(A) = 0$ and $\text{Tr}(B) = 2$.