I have a positive definite matrix of the form $Q+sI\alpha J$ ($s>2, 0 < \alpha <1$ and $J$ is the allones 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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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 $Q+sI$. 

