Updating the inverse of a matrix

Thanks to rank-one updates, we can bring that cost down to .At each step if is the current design matrix, our new datapoint is , then the updated version of is: and so that we have a rank-one update to for each new datapoint. Then, we have changed some elements of A, and we need to invert the matrix modified as follows.Of course, this problem can generally be solved by inverting the modified matrix.I am computing $(k I A)^$ in an iterative algorithm where $k$ changes in each iteration.

In particular, we find a way to compute the fundamental reproductive ratio of a relapsing disease being spread by a vector among two species of host that undergo a different number of relapses.

Now, my question is if there is an efficient way to compute the inverse that does not involve computing the inverse of a full $n$-by-$n$ matrix? I'll just write it here if someone else has need for it: $(k I A)^=(k I PDP^)^=(P(D k I)P^)^=P(D k I)^P^$, where $A=PDP^$ is the eigenvalue decomposition.

And the inverse of a diagonal matrix is quickly computed as the matrix with diagonal elements the reciprocal of the diagonal elements of the original matrix.

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