# Cholesky decomposition of $I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$

I need to compute the Cholesky decomposition of the following matrix:

$\varPi=I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$

Here $n$ is the dimension of the matrix and $x>0$. $\iota_{n}$ is an $n\times1$ column vector. The matrix is guaranteed to be positive definite.

Due to this simple structure is there anyway to obtain an analytical expression for the Cholesky decomposition or at least simplify the computational burden?

• It is not sure at all that a "simple structure" of a sym. Pos. Def. matrix yields analytical/explicit expressions for the coefficients of its Cholesky decomposition. Could you explain why are you interested in this particular form of matrix ? It may give a hint... – Jean Marie Jul 20 '16 at 14:42