# How does one implement the divide-and-conquer eigenvalue algorithm for a complex Hermitian matrix?

Let $T$ be a real symmetric tridiagonal matrix. Then the divide-and-conquer eigenvalue algorithm, as detailed in any standard text, goes by subdividing

$$T = \begin{bmatrix} T_1 & 0 \\ 0 & T_2 \end{bmatrix} + \rho vv^T,$$ after which we can solve the eigenvalue problem for $T_1, T_2$, and then solve the full eigenvalue problem by accounting for the rank-$1$ shift $\rho vv^T$ using a root-finding technique.

However, now let $T$ be a complex Hermitian tridiagonal matrix. Then if we attempt to do the same strategy, $$T = \begin{bmatrix} T_1 & 0 \\ 0 & T_2 \end{bmatrix} + S,$$ where $S$ is Hermitian but is not rank-$1$ (it appears to have rank $2$), so it seems like the same root-finding technique will fail, unless I'm mistaken. How can we adapt the algorithm to work with the complex Hermitian case?