▁→ ▁V ▁▁ ▁from ▁some ▁vector ▁space ▁V ▁to ▁itself , ▁is ▁a ▁sub space ▁W ▁of ▁V ▁that ▁is ▁preserved ▁by ▁T ; ▁that ▁is , ▁T ( W ) ▁ ⊆ ▁W . ▁ ▁General ▁description ▁▁ ▁Consider ▁a ▁linear ▁mapping ▁ ▁that ▁transform s : ▁ ▁An ▁invariant ▁sub space ▁▁ ▁of ▁ ▁has ▁the ▁property ▁that ▁all ▁vectors ▁ ▁are ▁transformed ▁by ▁ ▁into ▁vectors ▁also ▁contained ▁in ▁. ▁This ▁can ▁be ▁stated ▁as ▁ ▁T rivial ▁examples ▁of ▁invariant ▁sub spaces ▁ ▁: ▁Since ▁ ▁maps ▁every ▁vector ▁in ▁▁ ▁into ▁▁ ▁: ▁Since ▁a ▁linear ▁map ▁has ▁to ▁map ▁ ▁Un i - dimensional ▁invariant ▁sub space ▁U ▁The ▁basis ▁of ▁this ▁un i - dimensional ▁space ▁is ▁simply ▁a ▁vector ▁. ▁Con sequently , ▁any ▁vector ▁ ▁can ▁be ▁represented ▁as ▁ ▁where ▁ ▁is ▁a ▁real ▁scalar . ▁If ▁we ▁represent ▁ ▁by ▁a ▁matrix ▁ ▁then , ▁for ▁ ▁to ▁be ▁an ▁invariant ▁sub space ▁it ▁must ▁satisfy ▁ ▁We ▁know ▁that ▁ ▁with ▁. ▁ ▁Therefore , ▁the ▁condition ▁for ▁existence ▁of ▁a ▁un i - dimensional ▁invariant ▁sub space ▁is ▁expressed ▁as : ▁, ▁where ▁ ▁is ▁a ▁scalar ▁( in ▁the ▁field ▁of ▁the ▁vector ▁space ▁). ▁ ▁Note ▁that ▁this ▁is ▁the ▁typical ▁form ulation ▁of ▁an ▁eigen value ▁problem , ▁which ▁means ▁that ▁any ▁eigen vector ▁of ▁ ▁forms ▁a ▁un i - dimensional ▁invariant ▁sub space ▁in ▁ ▁F ormal ▁description ▁▁ ▁An ▁invariant ▁sub space ▁of ▁a