We recall that if $A$ is an invertible and symmetric real $m\times m$ matrix, then it is diagonalisable. Therefore, if we denote by $\mathcal{E}^{+}(A)\subset \mathbb{R}^m$ (respectively $\mathcal{E}^{-}(A)\subset \mathbb{R}^m$) to be the vector subspace generated by eigenvectors with positive eigenvalues of $A$ (correspondingly the vector subspace generated by eigenvectors with negative eigenvalues of $A$), then we have an orthogonal decomposition $\mathbb{R}^m=\mathcal{E}^{+}(A)\oplus \mathcal{E}^{-}(A)$... (read more)

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