▁eigenvalues , ▁since ▁distinct ▁eigenvalues ▁always ▁have ▁linear ly ▁independent ▁eigen ve ctors . ▁ ▁In ▁particular , ▁a ▁defect ive ▁matrix ▁has ▁one ▁or ▁more ▁eigenvalues ▁ λ ▁with ▁algebraic ▁multip licity ▁m ▁> ▁ 1 ▁( that ▁is , ▁they ▁are ▁multiple ▁roots ▁of ▁the ▁characteristic ▁polynomial ), ▁but ▁fewer ▁than ▁m ▁linear ly ▁independent ▁eigen ve ctors ▁associated ▁with ▁ λ . ▁If ▁the ▁algebraic ▁multip licity ▁of ▁ λ ▁exceed s ▁its ▁geometric ▁multip licity ▁( that ▁is , ▁the ▁number ▁of ▁linear ly ▁independent ▁eigen ve ctors ▁associated ▁with ▁ λ ), ▁then ▁ λ ▁is ▁said ▁to ▁be ▁a ▁defect ive ▁eigen value . ▁ ▁However , ▁every ▁eigen value ▁with ▁algebraic ▁multip licity ▁m ▁always ▁has ▁m ▁linear ly ▁independent ▁generalized ▁eigen ve ctors . ▁ ▁A ▁Herm it ian ▁matrix ▁( or ▁the ▁special ▁case ▁of ▁a ▁real ▁symmetric ▁matrix ) ▁or ▁a ▁unit ary ▁matrix ▁is ▁never ▁defect ive ; ▁more ▁generally , ▁a ▁normal ▁matrix ▁( which ▁includes ▁Herm it ian ▁and ▁unit ary ▁as ▁special ▁cases ) ▁is ▁never ▁defect ive . ▁ ▁Jordan ▁block ▁ ▁Any ▁n ont rivial ▁Jordan ▁block ▁of ▁size ▁ 2 × 2 ▁or ▁larger ▁( that ▁is , ▁not ▁completely ▁diagonal ) ▁is ▁defect ive . ▁ ▁( A ▁diagonal ▁matrix ▁is ▁a ▁special ▁case ▁of ▁the ▁Jordan ▁normal ▁form ▁and ▁is ▁not ▁defect ive .) ▁ ▁For ▁example , ▁the ▁n ▁× ▁n ▁Jordan ▁block , ▁ ▁has ▁an ▁eigen value , ▁ λ ,