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I've read this wonderful explanation of SVD, where the writer mentions that the columns of $V$ are the principal directions (Summary, #1). Is this also true when the data matrix $X$ is complex?

If I'm not mistaken, when $X$ is complex SVD gets us $X=USV^H$, where $(\cdot)^H$ stands for the conjugate transpose. I tried writing a few scripts, and I seem to be getting that the columns of $\bar{V}$ (conjugate of $V$) are in fact the principal directions. Is that wrong?

The reasoning behind this is that I was trying to use the directions as a base to transform the data. In such case, for example like in a Spherical Harmonics transform, one would obtain the coefficients via $w=Y^HX$, where $Y$ contains the directions (base). I manage to fit SVD notation to this notation by assuming that $\bar{V}$ are the directions, but not $V$.

Rodrigo de Azevedo
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Lior
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  • Isn't $V$ supposed to be a Hermitean matrix with $VV^{*}=\mathbb{I}$? In that case, where $v$ is the first column of $V$, we have $Xv=USV^{*}v=\sigma_1$ times the first column of $U$--and doesn't that make $v$, not $\bar v,$ a principal direction? If not, could you share (a) your definition of "principal direction" and (b) the evidence you have seen suggesting conjugation is required? – whuber Sep 23 '20 at 18:27
  • Indeed $V$ is a Hermitean matrix. – Lior Sep 24 '20 at 17:58
  • I guess the terminology I used is incorrect. I go by the same common definition of "principal direction" and I understand that the explanation provided indeed shows that the columns of $V$ are the directions. Let me rephrase: per the definition of SVD, for representing a row of data as a linear combination of the columns of $V$ one would calculate $w=XV$ to get the coefficients $w$. When performing a transformation, such as Spherical Harmonics transformation, one calculates $w=V^{-1}X$. Are these expressions identical and should they be? – Lior Sep 24 '20 at 18:05
  • $XV$ and $V^{-1}X$ are rarely identical; indeed, one of these expressions is nonsensical unless $X$ is a square matrix! – whuber Sep 24 '20 at 18:08
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    Alright, I think now it makes more sense to me. Many thanks! – Lior Sep 26 '20 at 07:28

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