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Suppose I perform dimension reduction using PCA:

dr = DimensionReduction[1, 2, 3, 2, 3, 5, 3, 5, 8, 4, 5, 
 8.5, Method -> "PrincipalComponentsAnalysis"]

If I want to see the principal components themselves, in the original space, one thought is to use the "OriginalData" feature of the DimensionReducerFunction, on basis vectors in the new space:

In[8]:= dr[1.0, 0.0, "OriginalData"]
dr[0.0, 1.0, "OriginalData"]

Out[8]= 1.86006, 2.9998, 4.81724

Out[9]= 3.38701, 3.01026, 5.64163

Is this a reasonable thing to do, or am I misinterpreting how the "OriginalData" feature works? And is there a better way to pull out the principal components themselves? People often want to visualize these for various reasons.

(There are several other questions about how to solve a similar problem with the PrincipalComponents function; this is a question about a different function.)

Here's your data:

data = 1, 2, 3, 2, 3, 5, 3, 5, 8, 4, 5, 8.5;
dr = DimensionReduction[data, Method -> "PrincipalComponentsAnalysis"];

This is not exactly a top level solution, but we can pick apart the DimensionReducerFunction and see inside. In there we have a matrix:

Transpose[dr[[1, "Model", "Matrix"]]]

-0.572383, -0.577502, -0.582125, 0.793367, -0.56945, -0.215163

I think these are the components. We can try to verify:

Transpose[Last[SingularValueDecomposition[2.0/Sqrt[3] Standardize[data], 2]]]

-0.572383, -0.577502, -0.582125, 0.793367, -0.56945, -0.215163

(I think we need the 2.0/Sqrt[3] because DimensionReduction appears to use biased standard deviation whereas Standardize uses the unbiased form. I'm not sure though.)