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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?

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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

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