Question: Principal Components Analysis

Let’s say you have analyzed a trivariate data base (three continuous variables) and have the following correlation matrix:

rij = 1.00 0.50 0.30
0.50 1.00 0.20
0.30 0.20 1.00

A. The two largest eigenvalues of | r – λI | = 0 are 1.68 and 0.83. What is the third value equal to? In general, how do we interpret eigenvalues?

B. The first two components are:

0.84 0.17
0.79 0.42

0.60 -0.79

Are these elements correlations or direction cosines? What does this mean exactly?

C. What are the third components – the actual figures, or loadings, for this third eigenvector? [Hint: the signs of the elements of this third column are: (going down) -, +, +].

D. Finally, now that you have the full matrix, what is the product of this matrix pre-multiplied by its transpose? Also, what is the product of the matrix post-multiplied by its transpose? What patterns do you see in these matrix products?

E. What is the "Component Score Coefficient Matrix?" (SPSS calls it this). And also, address these questions: for what purpose are these coefficients used? And, why are they scaled this way?

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