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I want to know, since the covariance matrix is symmetric, positive, and semi-definite, then if I calculate its eigenvectors what would be the properties of the space constructed by those eigenvectors (corresponds to non-close-zero eigenvalues), is it orthogonal or anything else special? Suppose this eigenvector matrix is called U, then what would be the properties with U*transpose(U)?

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    @Mitch: I tend to think of those things geometrically, much like Qiaochu's answer [here](http://math.stackexchange.com/questions/9758/9763#9763)... or you had something else in mind?2011-04-17

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A symmetric matrix has orthogonal eigenvectors (irrespective of being positive definite - or zero eigenvalues). Hence, if we normalize the eigenvectors, U * transpose(U) = I

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    @whuber: You're right.2011-04-17
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The eigenvectors correspond to the principal components and the eigenvalues correspond to the variance explained by the principal components.