Charlemagne Distinguished Lecture Series: Prof. Gilbert Strang
Prof. Dr. Gilbert Strang - Banded Matrices and Fast Inverses
Mathematics, Massachusetts Institute of Technology
The inverse of a banded matrix A has a special form with low rank submatrices except at the main diagonal. That form comes directly from the "Nullity Theorem." Then the inverse of that matrix A-1 is the original A - which can be found by a remarkable "local" inverse formula. This formula uses only the banded part of A-1 and it oers a very fast algorithm to produce A.
That fast algorithm has a potentially valuable application. Start now with a banded matrix B. (Possibly B is the positive denite beginning of a covariance matrix C - but covariances outside the band are unknown or too expensive to compute). It is a poor idea to assume that those unknown covariances are zero. Much better to complete B to C by a rule of maximum entropy - for Gaussians this rule means maximum determinant.
As statisticians and also linear algebraists discovered, the optimally completed matrix C is the inverse of a banded matrix. Best of all, the matrix actually needed in computations is that banded C-1 (which is not B !). And C-1 comes quickly and eciently from the local inverse formula.
A very special subset of banded matrices contains those whose inverses are also banded. These arise in studying orthogonal polynomials and also in wavelet theory|the wavelet transform and its inverse are both banded ( = FIR filters). We describe a factorization for all banded matrices that have banded inverses.