# Charlemagne Distinguished Lecture Series

### Monday, Nov 10, 2014 03:00 pm

Banded Matrices and Fast Inverses

Prof. Gilbert Strang (Massachusetts Institute of Technology)

**Abstract:**

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 o ers 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 de nite 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.

**Time**: 03:00-04:00 pm

**Location**: Ford-Saal, SuperC, RWTH Aachen, Templergraben 57, 52062 Aachen