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I'm trying to implement PCA biplot.

I've got much of the concepts nailed down specially thanks to:

Relationship between SVD and PCA. How to use SVD to perform PCA?

and

Positioning the arrows on a PCA biplot

There is one last detail I just can't wrap my head around.

Given:

$X=USV^t$

I should be able to get the coordinates of my variable vectors from the columns of $V$

What I dont understand is to what variable each coordinate comes from.

X is a n sample by p variable matrix, but V is a nxn matrix

kjetil b halvorsen
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jregalad
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  • $V$ cannot possibly be an $n\times n$ matrix, because that would give $X$ $n$ columns. $V$ usually is a $p\times p$ matrix. When you compute fewer than $p$ PCs, such as $k\lt p$ of them, $V$ is a $p\times k$ matrix. – whuber Jul 06 '17 at 17:58
  • I'm more confused now. So the V, the right matrix should always have the same number of columns than X. – jregalad Jul 06 '17 at 18:22
  • These are the rules of matrix multiplication: $X$ and $U$ *must* have the same numbers of columns; $X$ and $V^\prime$ *must* have the same numbers of rows. – whuber Jul 06 '17 at 18:45
  • Biplot and its relationship to PCA is explained also here quite comprehensively: https://stats.stackexchange.com/q/141754/3277 – ttnphns Jul 06 '17 at 19:15
  • @whuber, Thanks a lot. One las thing. Each entry of the columns of V correspond to a coordinate of my biplots. To what variable do they correspond then? – jregalad Jul 06 '17 at 21:31
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    The ith entry of the columns of V corresponds to the ith column in X – jregalad Jul 07 '17 at 08:13

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