▁... , ▁v k } ▁of ▁W ▁and ▁complete ▁it ▁to ▁a ▁basis ▁B ▁of ▁V . ▁Then , ▁with ▁respect ▁to ▁this ▁basis , ▁the ▁matrix ▁representation ▁of ▁T ▁takes ▁the ▁form : ▁ ▁where ▁the ▁upper - left ▁block ▁T 1 1 ▁is ▁the ▁restriction ▁of ▁T ▁to ▁W . ▁ ▁In ▁other ▁words , ▁given ▁an ▁invariant ▁sub space ▁W ▁of ▁T , ▁V ▁can ▁be ▁decom posed ▁into ▁the ▁direct ▁sum ▁ ▁View ing ▁T ▁as ▁an ▁operator ▁matrix ▁ ▁it ▁is ▁clear ▁that ▁T 2 1 : ▁W ▁→ ▁W ' ▁ ▁must ▁be ▁zero . ▁ ▁Det erm ining ▁whether ▁a ▁given ▁sub space ▁W ▁is ▁invariant ▁under ▁T ▁is ▁ost ens ibly ▁a ▁problem ▁of ▁geometric ▁nature . ▁Matrix ▁representation ▁allows ▁one ▁to ▁phrase ▁this ▁problem ▁algebra ically . ▁The ▁projection ▁operator ▁P ▁onto ▁W ▁is ▁defined ▁by ▁P ( w ▁+ ▁w ') ▁= ▁w , ▁where ▁w ▁ ∈ ▁W ▁and ▁w ' ▁ ∈ ▁W . ▁The ▁projection ▁P ▁has ▁matrix ▁representation ▁ ▁A ▁straightforward ▁calculation ▁shows ▁that ▁W ▁= ▁Ran ▁P , ▁the ▁range ▁of ▁P , ▁is ▁invariant ▁under ▁T ▁if ▁and ▁only ▁if ▁P TP ▁= ▁T P . ▁In ▁other ▁words , ▁a ▁sub space ▁W ▁being ▁an ▁element ▁of ▁Lat ( T ) ▁is ▁equivalent ▁to ▁the ▁corresponding ▁projection ▁satisfying ▁the ▁relation ▁P TP ▁= ▁T P . ▁ ▁If ▁P ▁is ▁a ▁projection ▁( i . e . ▁P 2 ▁= ▁P ), ▁so ▁is ▁ 1