# Cholesky decomposition of $A+kI$ given Cholesky decomposition of A

Suppose I have the Cholesky decomposition for a symmetric matrix $A$:

$$A = L L^T$$

I wish to compute the Cholesky decomposition for $A+kI$ where $I$ is the identity and $k$ is a scalar. Is there a way to obtain this using the decomposition for $A$ faster than recomputing the Cholesky decomposition from scratch?

• Have updated the question. If $k$ is negative then $A+kI$ may or may not be positive definite. For an example that is positive definite, take $A=2I$ and $k=-1$. – Alex Flint Jul 16 '13 at 19:45
• There is a literature on the low-rank update of Cholesky-factored matrices, of which this is a very narrow case. See e.g. citeseerx.ist.psu.edu/viewdoc/… – Nimrod Mar 9 '16 at 21:41