Sample Variance in Principal Components AnalysisI was reading this Why is the eigenvector of a covariance matrix equal to a principal component?. And in the top answer, the poster mentions that if the covariance matrix of the original data points x i was ∑, the variance of the new data points is just u T Σ u. I'm not sure why this is the case. Could someone enlighten me?

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Sample Variance in Principal Components AnalysisI was reading this Why is the eigenvector of a covariance matrix equal to a principal component?. And in the top answer, the poster mentions that if the covariance matrix of the original data points       x    i   was   ∑, the variance of the new data points is just       u    T        Σ    u. I&#039;m not sure why this is the case. Could someone enlighten me?

zepplinkid7yk · Accepted Answer

I Suppose your data set consists of N vectors, each of length T. Let&#039;s denote them                     x        →              i  , (i=1, ..., N). The new data set consists of                     y        →              i    =      ∑          k      =      1        N        p          i      k                          x        →              k  Then the covariance matrix of the new data set has elements                              C                      i            j                                              =                                          1                          T                              2                                  /                                α                                                                                        y              →                                i                ⋅                                            y              →                                j                                                  =                                          1                          T                              2                                  /                                α                                                    (                  ∑                      k            =            1                    N                          p                      i            k                                                              x              →                                k                )        ⋅        (                  ∑                      l            =            1                    N                          p                      j            l                                                              x              →                                l                )                                          =                              ∑                      k            =            1                    N                          ∑                      l            =            1                    N                          p                      i            k                                    p                      j            l                                    C                      k            l                    ′                                    C                            =                    P                  C          ′                          P          T                    where P is the matrix with components       p          i      k       and C&#039; is the covariance matrix of                     x        →              i  .

Sample Variance in Principal Components Analysis I was reading this Why is the eigenvector of a cov

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