▁shown ▁in ▁ 1 9 8 0 ▁by ▁Gol ub ▁and ▁Van ▁Lo an , ▁the ▁TLS ▁problem ▁does ▁not ▁have ▁a ▁solution ▁in ▁general . ▁The ▁following ▁cons iders ▁the ▁simple ▁case ▁where ▁a ▁unique ▁solution ▁exists ▁without ▁making ▁any ▁particular ▁assumptions . ▁ ▁The ▁computation ▁of ▁the ▁TLS ▁using ▁singular ▁value ▁decomposition ▁is ▁described ▁in ▁standard ▁texts . ▁We ▁can ▁solve ▁the ▁equation ▁▁ ▁for ▁B ▁where ▁X ▁is ▁m - by - n ▁and ▁Y ▁is ▁m - by - k . ▁▁ ▁That ▁is , ▁we ▁seek ▁to ▁find ▁B ▁that ▁minim izes ▁error ▁matrices ▁E ▁and ▁F ▁for ▁X ▁and ▁Y ▁respectively . ▁That ▁is , ▁ ▁where ▁ ▁is ▁the ▁augment ed ▁matrix ▁with ▁E ▁and ▁F ▁side ▁by ▁side ▁and ▁ ▁is ▁the ▁Fro ben ius ▁norm , ▁the ▁square ▁root ▁of ▁the ▁sum ▁of ▁the ▁squares ▁of ▁all ▁entries ▁in ▁a ▁matrix ▁and ▁so ▁equival ently ▁the ▁square ▁root ▁of ▁the ▁sum ▁of ▁squares ▁of ▁the ▁lengths ▁of ▁the ▁rows ▁or ▁columns ▁of ▁the ▁matrix . ▁ ▁This ▁can ▁be ▁re written ▁as ▁ ▁where ▁ ▁is ▁the ▁ ▁identity ▁matrix . ▁The ▁goal ▁is ▁then ▁to ▁find ▁ ▁that ▁reduces ▁the ▁rank ▁of ▁ ▁by ▁k . ▁Define ▁ ▁to ▁be ▁the ▁singular ▁value ▁decomposition ▁of ▁the ▁augment ed ▁matrix ▁. ▁ ▁where ▁V ▁is ▁partition ed ▁into ▁blocks ▁corresponding ▁to ▁the ▁shape ▁of ▁X ▁and ▁Y . ▁ ▁Using ▁the ▁E ck art – You ng ▁theorem , ▁the ▁approximation ▁minim ising ▁the