Kovarianzmatrix
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title: Covariance matrix
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content: 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.
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description: Verallgemeinerung der Varianz auf mehrdimensionale Zufallsvariablen @de
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description: Objektu-multzo bati p aldagai behatuz gero, bi aldagaikako kobariantzek osatzen duten (p, p) matrizea. Matrizearen diagonalean, beraz, aldagaien beren buruarekiko sakabanatzeak adierazten duten bariantzak gertatzen dira. @eu
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title: Proof
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description: ベクトル要素間の共分散の行列 @ja
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description: matice kovariancí náhodných veličin @cs
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