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Principal Component Analysis A newbie question -originally posted in un related forum

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Junior Member level 2
Mar 28, 2011
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Hi guys,

I'm trying to learn PCA. as an example I have just 2 points in my 3-D space. using PCA(eigen vectors) the outcomes are the principal component: line connecting these 2 points
the 2 other orthogonal axes in my opinion could be anywhere in the plane perpendicular to this connecting i wonder what is the PCA criteria to give me these 2 axes?
To make it clearer, please look at my Matlab example:

A= [1,2,3;1,8,5]; %My 2 points
[V, D]=eig(cov(A)) % Eigne vectors
V =
1.0000 0 0
0 0.3162 -0.9487
0 -0.9487 -0.3162
D =

0 0 0
0 0.0000 0
0 0 20.0000

I decided to choose another 2 vectors in the same plane instead of [0,0.3162,-0.9487 ] and
[0,-0.9487,-0.3162 ] .so i chose:

V2=[1,0,0;0,-0.6,0.8;0,0.8,0.6] % My orthogonal vectors:
new_cov2=cov(A*V2); % check how much variance,covariance I have in my data,using my vectors
new_cov2 =
0 0 0
0 2.0000 -6.0000
0 -6.0000 18.0000

does PCA try to Maximize variance in one axe and minimize in 2 other axes in the case of 2 points? i see 18 which is smaller than 20 and 2 which is larger than ~0.

Apreciate any insight regarding how PCA chooses 2 other axes in the case of having 2 points in the space of 3-D

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