I am reading about PCA and found an exercise that saysShow that when a N-dim set of data points X is projected onto the eigenvectors V = [ e 1 e 2 . . . e n ] of its covariance matrix C = X X T , the covariance matrix of the projected data C p = Y Y T is diagonal and hence that, in the space of the eigenvector decomposition, the distribution of X is uncorrelated.What I have so far is Y = V T XTherefore C p = Y Y T = V T X ( V T X ) T = V T X X T Vbut there I got stuck. Any advise on how to proceed, moreover, what does "The covariance matrix of the projected data is diagonal" mean?

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I am reading about PCA and found an exercise that saysShow that when a N-dim set of data points X is projected onto the eigenvectors   V  =  [      e    1        e    2    .  .  .      e    n    ] of its covariance matrix   C  =  X      X    T  , the covariance matrix of the projected data       C    p    =  Y      Y    T   is diagonal and hence that, in the space of the eigenvector decomposition, the distribution of X is uncorrelated.What I have so far is  Y  =      V    T    XTherefore      C    p    =  Y      Y    T    =      V    T    X  (      V    T    X      )    T    =      V    T    X      X    T    Vbut there I got stuck. Any advise on how to proceed, moreover, what does &quot;The covariance matrix of the projected data is diagonal&quot; mean?

Hayley Mccarthy · Accepted Answer

A diagonal matrix is one with zero everywhere and the diagonal entries can be zero or non zero.Assuming the eigenvectors are normalized in magnitude   ‖      e    i    ‖  =  1                    ∵                             V        =        [                  e          1                          e          2                …                  e          n                ]         and         C                  e          i                =                  λ          i                          e          i                                                  ∴                             C        V        =        V                  [                                                                      λ                  1                                                            0                                            ⋯                                            0                                                                    0                                                              λ                  2                                                            ⋯                                            0                                                                    ⋮                                            ⋮                                            ⋱                                            ⋮                                                                    0                                            0                                            ⋯                                                              λ                  n                                                              ]                                                  ∵                                       C          p                =                  V          T                (        X                  X          T                )        V        =                  V          T                C        V                                          ∴                                       C          p                =                  V          T                V                  [                                                                      λ                  1                                                            0                                            ⋯                                            0                                                                    0                                                              λ                  2                                                            ⋯                                            0                                                                    ⋮                                            ⋮                                            ⋱                                            ⋮                                                                    0                                            0                                            ⋯                                                              λ                  n                                                              ]                                    ∵                                       V          T                V        =                  I                                           as eigenvecotrs of a symmetric matrix are orthogonal                                          ∴                                       C          p                =                  [                                                                      λ                  1                                                            0                                            ⋯                                            0                                                                    0                                                              λ                  2                                                            ⋯                                            0                                                                    ⋮                                            ⋮                                            ⋱                                            ⋮                                                                    0                                            0                                            ⋯                                                              λ                  n                                                              ]                                          A diagonal matrix with variance in each eigenvector direction

I am reading about PCA and found an exercise that says Show that when a N-dim set of data points X

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