Mosaic Ranks for the Inverses to Band Matrices

After reminding the de nition of mosaic ranks we estimate them from above for the inverses to band matrices The estimate grows logarithmically with the matrix size The result presented in this note should be compared with the well known descriptions of the inverses to band matrices as semisep arable matrices Our approach still excells in that it holds under much weaker assumptions Introduction To begin with we remind some de nitions and well known facts Let p q n We say that A aij nij is a p q band matrix if aij whenever i j p or i j q It is very important what comes along the extreme diagonals i j p and i j q If there is a nonzero entry on each of them then p and q are correctly de ned upper and lower bandwidth If all their entries are nonze roes then A is called a strict p q band matrix This work was supported in part by the Russian Fund of Basic Research and the Volkswagen Stiftung Denote by Up and Lq the spaces of upper and lower triangular matrices of order p and q respectively We say that A is a q p semiseparable matrix if rst A S U rankS q U Un q and second A R L rankR p L Ln p It is known for years that the inverses to band matrices still capture rather imposing structure We all are aware of the following beautiful well known and rediscovered several times theorem A nonsingular matrix is a strict p q band matrix if and only if its in verse is a q p semiseparable matrix The proof can be found for example in see also How ever in contrast to the de nition and proof given in we need not to assume additionally that the diagonals i j q in S and R and i j p also are entry wise di erent The above theorem can be proved without this In fact the proof can be very short and clear Suppose rst that A is a nonsingular q p semiseparable matrix Rewrite in the block form A u v u v U u v u v We conclude immediately that the blocks u and v are nonsingular The Schur complement to the block is u v U u v u v u v U It implies that U is nonsingular and A U Using the same arguments from we derive that A L which completes the proof of the only if part The idea goes back probably to D K Faddeev Concerning the if part set A a b U c and remember the Frobenius formulas with h b aU c it holds that A U ch U U chaU h haU Still one ought to recover from this that