Let us look at three of the questions I asked early in February, and answer two of them.

## First, what do we know? What have we done?

We assume that we have data X, with the variables in columns, as usual. In fact, we assume that the data is at least centered, and possibly standardized.

We compute the covariance matrix

,

then its eigendecomposition

,

where is the diaginal matrix of eigenvalues. We define the -weighted matrix

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Finally, we use A as a transition matrix to define new data Z:

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We discovered two things. **One, the matrix A is the cross covariance between Z and X:**

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I find this interesting, and I suspect that it would jump off the page at me out of either Harman or Jolliffe; that is, I suspect it is written there but it didn’t register.

**Two, we discovered that we could find a matrix Ar which is the cross covariance between Zc and Xs.** Read the rest of this entry »