(June 10: i have made 4 edits, all cosmetic. you may search on “edit:”)

Malinowski (edit: “Factor Analysis in Chemistry”, 3rd ed.) does a lot of things differently from what we’ve seen. Fortunately, his model is simple enough, although his notation is… different. His model is

X = R C,

and he calls R and C the row and column matrices respectively. He wants X to have more rows than columns, so he transposes if necessary; then he chooses C to have more columns than rows, and R will have more rows than columns. For starters, then, his X matrix looks like the usual design matrix for regression. (Incidentally, he didn’t call it X.)

He chooses , from the cut-down SVD. That is, I write the SVD of X as

,

where u and v are orthogonal and w is the same shape as X. But we know from the derivation and our experience with Davis that we may also write

,

where is square, diagonal, and invertible (it is a cut-down w), and and are the submatrices of u and v which are conformable with . (We’ll see all this shortly.) We have dropped the parts of u, w, and v which are not required for reproducing X. (I remind you that what we’ve lost is the orthogonality of the matrices u and v.)

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