A Local Inverse Formula and a Factorization

  • Gilbert StrangEmail author
  • Shev MacNamara


When a matrix has a banded inverse there is a remarkable formula that quickly computes that inverse, using only local information in the original matrix. This local inverse formula holds more generally, for matrices with sparsity patterns that are examples of chordal graphs or perfect eliminators. The formula has a long history going back at least as far as the completion problem for covariance matrices with missing data. Maximum entropy estimates, log-determinants, rank conditions, the Nullity Theorem and wavelets are all closely related, and the formula has found wide applications in machine learning and graphical models. We describe that local inverse and explain how it can be understood as a matrix factorization.



The authors gratefully acknowledge a grant from The Mathworks that made this work possible.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.University of Technology SydneyUltimoAustralia

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