for this section, i need – i want – to match davis’ signs and sizes. this is just temporary.

instead of my …

i need to change the sign of the 2nd column and drop the 3rd:

instead of my …

i need to change the sign of the 1st column and drop the last two:

having those, it’s quickest to recompute and from the definitions. recall that

then

= x

=

(yes, those are davis’ signs.) then we have

= x

=

(yes, those too are davis’ signs.)

he has two plots. one is the so-called Q-mode loadings, the two nonzero columns of davis’ :

it would be informative for me to draw the so-called R-mode scores at this time, the two nonzero columns of davis’ :

i have arranged for Mathematica® to use the same imagesize and aspect ratio. gee, this is the same picture, except for the scales. (in fact, the book drew the Q-mode loadings but labeled it R-mode scores. we see that it’s an understandable error.)

the other plot he drew was the R-mode loadings, the two nonzero columns of his :

and i know you know we need to plot the Q-mode scores right now, the two nonzero columns of :

again, the scales have changed, but this is the same as the previous plot.

writing a little carelessly (exactly what do i mean by ?), the algebraic statements of what i have just shown you are :

hang on a second. suppose we compute and ; these two equations say that we can compute from and from . but computing directly gave us an eigenvector matrix U; computing directly gave us an eigenvector matrix V. these two equations say that we could have gotten without ever computing V.

if we have so many observations that V is huge, this is a useful trick.

in addition, however, it leaves one wondering what’s to be gained from Q-mode analysis. we have not established that there’s nothing to be gained; but we can get whatever it is, if anything, without ever forming . (and yes, there is something to be gained. believe me!)

to confirm those relationships among the cut-down matrices, we need a 2×2 matrix instead of a 3×3 or 4×4 namely:

let’s confirm . the RHS is:

x

=

and the LHS is:

which is what we wanted to see.

similarly, for , the RHS is

x

=

and the LHS is

as it should be.

so: if we cut down the matrices and we use davis’ signs, we see that the Q-mode FA could have been computed from the R-mode..

next, we will look at that using the SVD.

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