Question: Eigenvalues, eigenvectors

[Multivariate distributions] Let \(A\) be a symmetric matrix (such a matrix appears when considering multivariate distributions).

1. Show that the eigenvalues of \(A\) are real.
2. Assume that \(A=\left(\begin{array}{ccc}1.5 & 0.5 & 0 \\ 0.5 & 1.5 & 0 \\ 0 & 0 & 3\end{array}\right)\). Compute the eigenvalues and
   eigenvectors of \(A\).
3. Find a matrix \(S\) and a diagonal matrix \(D\) such that \(A=S D S^{-1} .\) Can you choose \(S\) so that \(S^{-1}=S^{\prime}\) (where \(S^{\prime}\) is the transpose of \(\left.S\right)\) ?

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