# Eigenvalues relationship

I have $n \times n$ real matrices $A$ and $D$. $D$ is diagonal. Let's $v_i(A), \lambda_i(A)$ be a couple of eigenvectors-eigenvalues of $A$. What relationships there exists between $v_i(B), \lambda_i(B)$ and $v_i(A), \lambda_i(A)$ where $B = DA$?

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Example in $\def\R{\Bbb R}\R^2$. With $A=(\begin{smallmatrix}0&1\\1&0\end{smallmatrix})$ one has eigenvalues $+1,-1$ with eigenvectors that you can easily spot. Multiplying by $D=(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix})$ one gets $B=DA=(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix})$, without any real eigenvalues at all. ($B$ has complex eigenvalues $\pm\mathbf i$, with eigenvectors that bear no relation to those of $A$).