Schur decomposition of matrix $X - \alpha \, u_1 u_1^T$

Question

Consider the Schur decomposition $X = URU^T$ of a real matrix $X$, where $U$ is orthogonal and $R$ is upper triangular.

Is there a nice way to compute the Schur decomposition of the matrix $X - \alpha \, u_1 u_1^T$ for any $\alpha \in \mathbb{R}$, where $u_1u_1^T$ is the outer product of the first column of $U$ with itself?

I'm not sure how to use the decomposition of $X$ here.

Answer 1

The same orthogonal matrix $U$ will give the Schur decomposition:

$$U^\top(X-\alpha u_1u_1^\top)U=U^\top XU-\alpha U^\top u_1u_1^\top U=R-\alpha e_1e_1^\top=\tilde{R}$$ $$\therefore\ X-\alpha u_1u_1^\top=U\tilde{R}U^\top$$ where $\tilde{R}$ is the same as matrix $R$ except $\tilde{R}_{11}=R_{11}-\alpha$. Note that $U^\top u_1=U^{-1}u_1=e_1$.