I have a problem involving a complex matrix that I need to diagonalize and apply weights to the entries according to a function relating only to the eigenvalues of the real part of the matrix.
Suppose that I have diagonalized separately the real and imaginary parts of $ A=A_r+iA_i $ such that: $ A_r=U_r\Lambda_rU_r^{-1} $ and $ A_i=U_i\Lambda_iU_i^{-1} $. What I need to do next is calculate a weighted "error" norm which, given two complex vectors $ x,\ y $ is written as:
$ \sum_{i=1}^N w_i|y-Ax|_i^2 $ and $ A=U_r\Lambda_rU_r^{-1}+iU_i\Lambda_iU_i^{-1} $
I then rewrite the problem as:
$ \begin{bmatrix} y_r \newline y_i \end{bmatrix}=\begin{bmatrix} A_r & -A_i \newline A_i & A_r \end{bmatrix} \cdot \begin{bmatrix} x_r \newline x_i \end{bmatrix} $
I thought that to correctly apply weights to my rows, I would first need to find a new $ 2N\times 2N $ orthogonal matrix $ U $ such that $ \tilde{y}=U^{-1}\begin{bmatrix} y_r \newline y_i \end{bmatrix},\ \tilde{x}=U^{-1}\begin{bmatrix} x_r \newline x_i \end{bmatrix} $ and $ \sum w_i|y-Ax|_i^2 = \sum w_i|\tilde{y}-\Lambda\tilde{x}|_i^2 $
where $ \Lambda $ would be a new diagonal matrix* and then I could correctly apply the weights to my rows according to some function of the $ [\Lambda_r]_{ii} $ for each $ i $. Essentially, is there any way to find a new diagonalization $ U,\ \Lambda $ such that:
$ \begin{bmatrix} U_r\Lambda_r U_r^{-1} & -U_i\Lambda_i U_i^{-1} \newline U_i\Lambda_i U_i^{-1} & U_r\Lambda_r U_r^{-1} \end{bmatrix} = U\Lambda U^{-1} $
Thanks for your inputs in advance!
*(possibly $ \begin{bmatrix} \Lambda_r & 0 \newline 0 & \Lambda_i \end{bmatrix} $ ?)