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!
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asked Apr 25, 2012 at 0:29
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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 $Q+sI$.
answered Apr 25, 2012 at 1:23
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$\begingroup$ Thanks! Can you give me a particular reference? My focus is on theory, not computation. $\endgroup$ Commented Apr 25, 2012 at 7:53
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$\begingroup$ 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 $\endgroup$ Commented Apr 26, 2012 at 1:02
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$\begingroup$ 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. $\endgroup$ Commented Apr 26, 2012 at 4:52
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$\begingroup$ Well, actually it's $\alpha$ that's varying and $s$ is fixed. Thanks for the reference, I'll be sure to read. $\endgroup$ Commented Apr 29, 2012 at 14:08