WE GIVE A SEMIDIRECT PRODUCT DECOMPOSITION FOR AN ABELIAN GROUPOID. THIS IS DONE THROUGH A MATCHED PAIR AND LEADS TO A C* -DIAGONAL (FOR A SPECIAL CASE). WE USE THIS DECOMPOSITION TO STUDY THE NORM-ONE EIGENVECTORS OF CORRESPONDING FULL C* -ALGEBRA INSTEAD MULTIPLICATIVE FUNCTIONALS (SPECTRUM) WHICH HAS NORM-ONE IN ABELIAN GROUP CASE TO DEFINE DUAL GROUP.