Check under ambidexterity
"Without this feat, a character who uses his or her off hand suffers a -4 penalty to attack rolls, ability checks, and skill checks. For example, a right-handed character wielding a weapon with her left hand suffers a -4 penalty to attack rolls with that weapon."
But no penalty to str bonus applied. <---- *sigh* This is incorrect, I spoke too quickly, I double checked and the Glossary of the PHB notes that the half-str bonus penalty applies.