scaling down the imarinary parts of eigenvalues of a matrix

Question

Let A be an n-by-n complex matrix. Is there a transformation to preserve the real parts of the eigenvalues of A but scale down the imaginary parts of the eigenvalues of A? Actually , I want to have a matrix that keeps the real parts of the eigenvalues of the original matrix but have the sum of absolute of the imaginary parts of the original eigenvalues minimized or reduced.

Answer 1

It depends what you mean by "transformation".

For instance, you can use the expression $$B=\dfrac{1}{2}P(J+\bar J)P^{-1}$$ where $A=PJP^{-1}$ and $J$ is the Jordan form of $A$ and $\bar J$ is the complex conjugate of $J$ (componentwise).

In this case, $B$ will have real eigenvalues which are equal to the real part of the eigenvalues of $A$.

An interesting point of this transformation is that it preserves the structure of the matrix by keeping its eigenvectors and its Jordan structure. Only the eigenvalues are changed.