# Principal components using correlation matrix in R

My understanding is that prcomp and princomp work off the dataset itself (row of observations, across variables in the columns). Is there a function that will run a principal component analysis directly off a correlation or covariance matrix, without having the "raw" dataset?

• You just may do singular-value or eigen-value decomposition of the matrix, to get eigenvalues and eigenvectors (and then compute loadings out of the two). But without casewise data you won't get component scores, of course. Jul 15 '14 at 20:39

You can use eigen(). For example:

> set.seed(3)
> x <- matrix(rnorm(18), ncol=3)
> x
[,1]        [,2]       [,3]
[1,]  1.2243136 -0.48445511  0.9006247
[2,]  0.1998116 -0.74107266  0.8517704
[3,] -0.5784837  1.16061578  0.7277152
[4,] -0.9423007  1.01206712  0.7365021
[5,] -0.2037282 -0.07207847 -0.3521296
[6,] -1.6664748 -1.13678230  0.7055155

> prcomp(x)
Standard deviations:
[1] 1.0294417 0.9046837 0.4672911

Rotation:
PC1         PC2         PC3
[1,] -0.84047203 -0.53902142  0.05534150
[2,]  0.53878561 -0.84219645 -0.02037687
[3,] -0.05759199 -0.01269102 -0.99825954

> eigen(cov(x))
$values [1] 1.0597501 0.8184527 0.2183610$vectors
[,1]       [,2]        [,3]
[1,]  0.84047203 0.53902142 -0.05534150
[2,] -0.53878561 0.84219645  0.02037687
[3,]  0.05759199 0.01269102  0.99825954


So the eigenvalues of the covariance matrix are the squares of the standard deviations (i.e, variances) of the principal components and the principal components themselves are same as eigenvectors of covariance matrix (though signs may be opposite as they are here).

• Thanks for providing the details. A bit of searching produced this link, where fa from the psych package is used: stats.stackexchange.com/questions/31948/… Jul 15 '14 at 22:19