So, it decomp3 gives me three variables.

D, the singular values.

U, the left single vectors.

V, the right singular vectors.

So just to show that these are all getting at the same thing.

Let me combine the eigan vectors from the eiganvalue decomposition

of the correlation matrix, and

I'll do it with are called the loadings from the printcomp function.

And then I'll do it,

I'll also grab what is the v matrix from the singular value decomposition.

And here are those three quantities.

Notice like, take for example this first column, Comp.1, Comp.2,

Comp.3, Comp.4, 0.524, 0.258, and so on.

Then if I go down to the output from printcomp, 0.524, 0.258, -0.003.

There's no reason, by the way, that the signs should have to agree in this case.

I think we're just fortunate that we did because they're using

underlying the same exact numerical libraries.

But there's no reason that there's a sign in variance to all these decompositions.

And then of course, the singular value decomposition gives us the same things,

and then if we go through the second row.

It's the same thing, again.

So, what we see is that, the three different approaches are giving us

the same v matrix, and then if I were to plot my eigenvalues.

Here's the first row are the eigenvalues from

the eigenvalues decomposition of the correlation matrix.

The second one are the eigenvalues from the printcomp function that what calls

those standard deviations, so we have to square them to get the eigenvalues.