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The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.

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Follow up: Show that these vectors are linearly independent almost surely

This is not an answer, just a slightly different perspective on the original problem. If I understand your setup correctly, you have two families of hypersurfaces: those of the form $E_i(P)=\{x\in \m …
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