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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


marked as duplicate by Jack M, user127.0.0.1, Claude Leibovici, Macavity, Stefan Hansen Mar 15 '14 at 6:47

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migrated from stats.stackexchange.com Mar 10 '14 at 10:37

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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 '14 at 10:31
  • $\begingroup$ See mathworld.wolfram.com/Eigenvalue.html which has a sentence about that. $\endgroup$ – Mok-Kong Shen Mar 10 '14 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 '14 at 12:20
  • $\begingroup$ Both answers given so far should be comments. I'm flagging both. $\endgroup$ – Git Gud Mar 10 '14 at 12:46

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.

  • $\begingroup$ 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 '14 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 '14 at 5:36
  • $\begingroup$ Done. math.stackexchange.com/questions/707855/… $\endgroup$ – Shashank Mar 11 '14 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. $\endgroup$ – Parag S. Chandakkar Jun 27 '16 at 14:19
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    $\begingroup$ @ParagS.Chandakkar - Thanks, fixed. $\endgroup$ – nbubis Jun 27 '16 at 14:21

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