# Finding a digonal matrix such D such that eigenvalues of B equals eigenvalues of A + D

by Jerry   Last Updated April 16, 2018 12:20 PM - source

There are two matrices $A,B$ $\in$ $\mathbb{R}^{n \times n}$; I know the eigenvalues of $A$ and $B$. Can I find a diagonal matrix $D =$ diag $\{d_1, d_2, \cdots, d_n\}$ such that the characteristic polynomial $\chi_B=\chi_{A+D}$?

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Knowledge of $spectrum(A),spectrum(B)$ is useless. Write the equality of the $2$ characteristic polynomials. You obtain a system of $n$ equations in the $n$ unknowns $(d_i)$. In the generic case, that reduces to solving a polynomial of degree $n!$. The previous polynomial has not necessarily any real solution.