Eigenvalues vs PVE (percent variance explained)

Question

With prcomp() function, I have estimated percent variance explained

prcomp(env, scale=TRUE)

The second column of summary(pca) shows these values for all PCs:

                        PC1    PC2     PC3     PC4     PC5     PC6     PC7
Standard deviation     7.3712 5.8731 2.04668 1.42385 1.13276 0.79209 0.74043
Proportion of Variance 0.5488 0.3484 0.04231 0.02048 0.01296 0.00634 0.00554
Cumulative Proportion  0.5488 0.8972 0.93956 0.96004 0.97300 0.97933 0.98487

Now I want to find what the Eigenvalues for each PC are:

pca$sdev^2
[1] 5.433409e+01 3.449329e+01 4.188887e+00 2.027337e+00 1.283144e+00
[6] 6.274083e-01 5.482343e-01

But these values appear to be simply an alternate representation of the PVE itself. So what am I doing wrong here?

Answer 1

I'm not sure if this is your confusion.

pca$sdev^2 -> eigen values -> variance in each direction
pca$sdev^2/sum(pca$sdev^2) = proportion of variance vector

So they ARE related.

Edit: Just an example (to illustrate this relationship), if that'll help.

set.seed(45) # for reproducibility
# set a matrix with each column sampled from a normal distribution
# with same mean but different variances
m <- matrix(c(rnorm(200,2, 10), rnorm(200,2,10), 
               rnorm(200,2,10), rnorm(200,2,10)), ncol=4)
pca <- prcomp(m)

> summary(pca) # note that the variances here equal that of input
# all columns are independent of each other, so each should explain
# equal amount of variance (which is the case here). all are ~ 25%
                           PC1     PC2     PC3    PC4
Standard deviation     10.9431 10.6003 10.1622 9.3200
Proportion of Variance  0.2836  0.2661  0.2446 0.2057
Cumulative Proportion   0.2836  0.5497  0.7943 1.0000

> pca$sdev^2
# [1] 119.75228 112.36574 103.27063  86.86322

> pca$sdev^2/sum(pca$sdev^2)
# [1] 0.2836039 0.2661107 0.2445712 0.2057142