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I have a square matrix that is largely zeros but interspersed with values. Is there a way to 'solve' this matrix so that all the information in it is contained in its lower triangle only, with the upper triangle containing only zeros?

L Fischman
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  • I'm voting to close this question as off-topic because it is about [math.se] instead of programming or software development. – Pang Aug 24 '18 at 06:11

1 Answers1

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Not in general.

  • If symmetric and positive definite you can do a Cholesky Decomposition.
  • If non-symmetric you can do an LU decomposition.
  • The Q matrix in quadratic forms (x'Qx) can be made symmetric and then lower-triangular. This is sometimes used when setting up a Quadratic Programming (QP) model.

Sparse versions of decomposition approaches are a well-studied area (not trivial though). In large-scale LP solvers sparse LU (simplex) or sparse Cholesky (interior point) are widely used.

Erwin Kalvelagen
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  • Thanks, I will try those. The matrix was sparse enough that what I did for the short term was just reorder the rows so that 'unknowns' did not have to rely on unknowns, unless the previous unknowns were first solved by reference to 'knowns'. Given this is a square matrix with many more knowns than unknowns, isn't it a given that an LP solver would work? – L Fischman Apr 02 '18 at 21:08
  • An LP solver can solve just a system of sparse linear equations. The presolver will often detect and remove a triangular part. Note that for good performance you will need to tell the solver that all variables are basic (advanced basis or warm start). This will allow the solver to solve the system of equations in 0 simplex iterations (just an invert). – Erwin Kalvelagen Apr 03 '18 at 02:08