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I'm running a PCA using prcomp in R and I get a table like this from the summary function: 

Importance of components:
                        PC1     PC2     PC3     PC4
Standard deviation     227.5998 86.2614 6.76700 3.29498
Proportion of Variance   0.8736  0.1255 0.00077 0.00018
Cumulative Proportion    0.8736  0.9990 0.99982 1.00000

Which gives me eigenvalues of:

 PC1      PC2      PC3      PC4 
51801.68  7441.03  45.79    10.86

My understanding was that any variable with eigenvalues greater than 1 are considered important but these seem ridiculously high. And I wouldn't think a principal component with a proportion of variance of 0.00018 to be too important. All my data seems fine.

Thanks!

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    $\begingroup$ A tell-tale sign of using variables as they come is that the eigenvalues don't sum to the number of variables (implying that the mean eigenvalue is 1). As the fine answer by @Mark L. Stone tells you, you didn't scale values first, so your first PC1 is just dominated by variable(s) with the highest variance(s) (which may be a side-effect of particular units of measurement). $\endgroup$
    – Nick Cox
    Commented Feb 26, 2018 at 21:07

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Apparently you used the default value of scale = FALSE, so any rule of thumb you have about eigenvalue importance as judged relative to a threshold of 1 would be invalid. The output is telling you the proportion of variance for each principal component.

I believe with default settings for tol and rank, all components are shown, regardless of importance.

See https://www.rdocumentation.org/packages/stats/versions/3.4.3/topics/prcomp

Also note that the eigenvalues are the squares of the standard deviations.

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    $\begingroup$ Take my answer plus the amplifying comment above from @Nick Cox, and I think that pretty much tells you what's going on. As the documentation in the link I provided says "The default is FALSE for consistency with S, but in general scaling is advisable." $\endgroup$
    – Mark L. Stone
    Commented Feb 27, 2018 at 1:06

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