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I would like to know if anyone knows any reference about rank one update of Schur decomposition.

Assume that we know the Schur decomposition of $A,$ which is $A=QUQ^{-1},$ where $Q$ is unitary and $U$ is upper-triangular.

Question 1: is there a fast way to compute the Schur decomposition of $B=A+uv^T,$ where $u,v$ are both rank-one?

Question 2: what if $B=A+\begin{bmatrix}\pmb{0} \\ \vdots \\ u \\ \vdots \\ \pmb0 \end{bmatrix},$ which means the rank-one update of $A$ is just basically a matrix with only one nonzero row?

I have read some reference and see that such as rank-one update for a QR decomposition can be done in $O(n^2)$ operations, where $n$ is the size of the matrix. I feel like this is also hopefully true for Schur decomposition. However, I tried and wasn't able to find related reference.

Any source / comment would be appreciated. Thank you!

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  • $\begingroup$ Corrected. Thanks for pointing out. $\endgroup$ Commented Jul 25 at 16:23

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