cholesky decomposition of special correlation matrix

Question

Suppose the covariance matrix is of the form $A = aI +bJ$, where $I$ is the identity matrix and $J$ is the matrix of all ones. Assume that Cholesky decomposition is possible. Do we get a nice analytic expression for Cholesky decomposition of $A$?

Answer 1

The Cholesky factorization of $aI$ is

$aI=R^{T}R$

where $R=\sqrt{a}I$. This is easy to compute.

The matrix of all ones, $J$ is a rank one matrix, so

$J=vv^{T}$

where

$v=\left[ \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \end{array} \right] $.

Thus

$aI+bJ= a + bvv^{T}$

This is a rank-one update (or "downdate" if $b<0$) of $aI$. There are simple and fast algorithms for the Cholesky factorization of a rank-one update to a matrix that can be used. In MATLAB this is implemented in the cholupdate() function.