I have a positive definite matrix of the form $Q+sI-\alpha J$ ($s>2, 0 < \alpha <1$ and $J$ is the all-ones matrix), where $Q$ is "nice", nonnegative and known. I'd like to know if there is a way to obtain an explicit expression for the Cholesky factorization of my matrix in this special case. Thanks!


The matrix $\alpha J$ is a rank one matrix, so there are simple update/downdate formulas for computing the Choleksy factorization of $Q+sI-\alpha J$ if you start with the factorization of $Q+sI$.

I'm not aware of any update formulas that get you from the Cholesky factorization of $Q$ to a Cholesky factorization of $Q+sI$.

| cite | improve this answer | |
  • $\begingroup$ Thanks! Can you give me a particular reference? My focus is on theory, not computation. $\endgroup$ – Felix Goldberg Apr 25 '12 at 7:53
  • $\begingroup$ A classic reference (and the paper is available online as a free .pdf) is: P. Gill, G. Golub, W. Murray, and M. Saunders. Methods for modifying matrix factorizations. Mathematics of Computation, 126(28):505-535, 1974. stanford.edu/group/SOL/papers/ggms74.pdf $\endgroup$ – Brian Borchers Apr 26 '12 at 1:02
  • $\begingroup$ Let me also mention that if $\alpha$ is fixed and you want to vary $s$, then you might find that a better way to go is to compute the eigenvalue decomposition of $Q-\alpha J$, and then adjust for $sI$ by adding $s$ to the eigenvalues. $\endgroup$ – Brian Borchers Apr 26 '12 at 4:52
  • $\begingroup$ Well, actually it's $\alpha$ that's varying and $s$ is fixed. Thanks for the reference, I'll be sure to read. $\endgroup$ – Felix Goldberg Apr 29 '12 at 14:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.