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- Indeed, from the property 4 it follows that under linear transformation of random variable with covariation matrix by linear operator s.a. , the covariation matrix is tranformed as
: .
As according to the property 3 matrix is symmetric, it can be diagonalized by a linear orthogonal transformation, i.e. there exists such orthogonal matrix , that
:
and are eigenvalues of . But this means that this matrix is a covariation matrix for a random variable , and the main diagonal of consists of variances of elements of vector. As variance is always non-negative, we conclude that for any . But this means that matrix is positive-semidefinite. (en)
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