how to construct a special matrix out of two lists

F[abs_, trs_] := ReplacePart[KroneckerProduct[trs, abs], {k_, k_} -> 0]

F[{a1, a2, a3, a4}, {t1, t2, t3, t4}] // MatrixForm

$\left( \begin{array}{cccc} 0 & \text{a2} \text{t1} & \text{a3} \text{t1} & \text{a4} \text{t1} \\ \text{a1} \text{t2} & 0 & \text{a3} \text{t2} & \text{a4} \text{t2} \\ \text{a1} \text{t3} & \text{a2} \text{t3} & 0 & \text{a4} \text{t3} \\ \text{a1} \text{t4} & \text{a2} \text{t4} & \text{a3} \text{t4} & 0 \end{array} \right)$

Two other alternatives

Outer[Times, trs, abs] - DiagonalMatrix[abs*trs]

Transpose[{trs}].{abs} - DiagonalMatrix[abs*trs]