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What exactly do eigenvalue and eigenvector indicate?

I mean what is the importance in calculating them. We all know how to calculate them but I do not know the exact importance

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  • $\begingroup$ Absent a specific statistical context, this is purely a mathematical question, and probably belongs on math.SE. $\endgroup$
    – Glen_b
    Mar 10, 2014 at 10:31
  • $\begingroup$ See mathworld.wolfram.com/Eigenvalue.html which has a sentence about that. $\endgroup$
    – Mok-Kong Shen
    Mar 10, 2014 at 10:54
  • $\begingroup$ As usual, the Wikipedia page is a good place to start. In case you overlooked it in the previous sentence, here is the link again: link. The link above will take you directly to applications of eigenvectors in PCA, which is what you specifically asked about. And the rest of that page contains several other applications. $\endgroup$
    – bubba
    Mar 10, 2014 at 12:20
  • $\begingroup$ Both answers given so far should be comments. I'm flagging both. $\endgroup$
    – Git Gud
    Mar 10, 2014 at 12:46

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Think of a matrix as an operator - one that operates on vectors, and returns another vector, which is a rotated, skewed and stretched copy of the original. Now, if we ignore length changes, some of the vectors will be left unchanged by this matrix, and retain their overall direction (though not their length): It is these vectors we call the eigenvectors of the matrix, and the factor by which their magnitude changes is called their eigenvalue.

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  • $\begingroup$ Also, can you please tell me why eigenvalues are used in PCA. Specifically why and how does it explain the variance of the components $\endgroup$
    – Shashank
    Mar 11, 2014 at 4:03
  • $\begingroup$ @Shashank - no. If you want to ask about a totally unrelated question, please open a new question on the topic. $\endgroup$
    – nbubis
    Mar 11, 2014 at 5:36
  • $\begingroup$ Done. math.stackexchange.com/questions/707855/… $\endgroup$
    – Shashank
    Mar 11, 2014 at 10:30
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    $\begingroup$ @nbubis Did you mean to say: It is these vectors we call the eigenvectors of the matrix. $\endgroup$
    – Autonomous
    Jun 27, 2016 at 14:19
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    $\begingroup$ @ParagS.Chandakkar - Thanks, fixed. $\endgroup$
    – nbubis
    Jun 27, 2016 at 14:21

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