$\boldsymbol A$ and $\boldsymbol B$ are $2\times 2$ matrices. $\boldsymbol 0$ refers to a $2\times 2$ zero matrix.
$\boldsymbol C$ is a block circulant matrix with $2n$ dimensions. $n$ is variable. For example, for $n=3$, $$ \boldsymbol C = \begin {bmatrix} \boldsymbol A & \boldsymbol B & \boldsymbol 0 \\ \boldsymbol 0 & \boldsymbol A & \boldsymbol B \\ \boldsymbol B & \boldsymbol 0 & \boldsymbol A \end {bmatrix} \text , $$ where $\boldsymbol C$ is a $6\times 6$ matrix. $\boldsymbol A$ and $\boldsymbol B$ are known. Some eigenvalues of $\boldsymbol C$ have positive real parts.
$\boldsymbol H$ is a diagonal matrix with one nonzero element, i.e., $\boldsymbol H=\operatorname {diag}(0,k,0,0,0,0)$, ($k<0$).
With increasing $n$ (the matrix dimensions), can we find any condition for the bound of $k$ to make all the eigenvalues of $ \boldsymbol C + \boldsymbol H $ in left-half plane?