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For a symmetric matrix $\mathbf{H}\in \mathbb{R}^{n\times n}$, we have a
decomposition 
\begin{equation*}
\mathbf{H=U\Lambda U^{\mathrm{T}}}
\end{equation*}
where the diagonal matrix $\mathbf{\Lambda} $ contains the eigenvalues $\lambda _{k}$
and $\mathbf{U}=(\mathbf{u}_1,\ldots ,\mathbf{u}_{n})$ contains the eigenvectors.

For $\mathbf{H}$ symmetric and real, the eigenvalues $\lambda_{k}$ are real and the
eigenvectors associated to different eigenvalues are orthogonal, that is $\mathbf{U}^{\mathrm{T}}
\mathbf{U} = \mathbf{I}$. 

Using the eigenvalue decomposition of the Hessian $f''
=\mathbf{U\Lambda U^{\mathrm{T}}}$, we obtain from the Taylor series~

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