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Assume one knows the eigenvalues $(\lambda_i)$ of a real matrix $M$ of size $n \times n$. Let $b$ be a vector in $\mathbb{R}^n$ and construct the following matrix $J$ by: $J_{ij} = M_{ij}b_j$ Can we deduce the eigenvalues of $J$ from the eigenvalues $\lambda_i$ and the vector $b$ ?

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    @mellow It was a little bit puzzling for me since $b$ is a vector. I see that you are not summing over indices.2011-10-10

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No. For example, suppose $n=2$ and we know that $M$ has eigenvalues $0$ and $2$ and that $b=(1,0)$. Then we might have $M=\begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}$, in which case $J=\begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}$ has eigenvalues $0$ and $2$. On the other hand, we might have $M=\begin{pmatrix} 0 & 0 \\ 0 & 2 \end{pmatrix}$, in which case $J=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ has $0$ as its only eigenvalue.