Note: This is a short version of https://math.stackexchange.com/questions/979834/
For a $n$-dimensional symmetric matrix A, orthogonal matrix C exists so that the relationship
$$AC = CD$$ is fulfilled, where $D$ is a diagonal matrix. For $D=\delta_{ij}d_j$ and $C=\begin{pmatrix}c_1 & \cdots & c_n\end{pmatrix}$ (i.e. $c_j$ are the column vectors of $C$ the eigenvalue equation
$$Ac_j = d_jc_j \Rightarrow (A-d_jE)c_j = 0$$
results, which are $n$ equations ($E$ is the identity matrix).
Now, if a constraint is given to $D$ (i.e. the eigenvalues) such as
$$D = \begin{pmatrix}m_1 d_1 & & \\ & \ddots & \\ & & m_n d_n\end{pmatrix}$$
it should result in $n$ equations with
$$(A-m_1d_1E)c_1 = 0$$
$$ \vdots$$
$$(A-m_n d_n E)c_n = 0$$
Instead, my textbook suggests
$$(A-d_j M)c_j = 0$$
where $M$ is a diagonal matrix with:
$$ M = \begin{pmatrix}m_1 & & \\ & \ddots & \\ & & m_n \end{pmatrix}$$
Which one is correct and why?