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I'd like to find the SVD for the following matrix: $$\left(\begin{array}{cc} 5 & 5\\ -1 & 7 \end{array}\right)$$

I've calculated the eigenvalues and eigenvectors of $XX^{T}$ and the eigenvectors of $X^{T}X$, arriving at $$X=\left(\begin{array}{cc} 5 & 5\\ -1 & 7 \end{array}\right)=U\Sigma V^{T}=\left(\begin{array}{cc} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{array}\right)\left(\begin{array}{cc} \sqrt{80} & 0\\ 0 & \sqrt{20} \end{array}\right)\left(\begin{array}{cc} \frac{1}{\sqrt{10}} & -\frac{3}{\sqrt{10}}\\ \frac{3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{array}\right)^{T}$$ This does not work. However if I multiply the second column of $U$ by -1: $$\left(\begin{array}{c} \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}} \end{array}\right)\mapsto\left(\begin{array}{c} -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} \end{array}\right)$$ Everything works out.

The question is why? Both are normalized orthogonal eigencectors, shouldn't either one work? How do I choose?

H.Rappeport
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  • A similar question was asked and answered here. Perhaps you'll find my answer there satisfactory. – Ben Grossmann Apr 17 '18 at 15:33
  • @Omnomnomnom. I somehow missed that one searching for an answer, and it indeed answers the question, thanks. Should I delete this one? – H.Rappeport Apr 17 '18 at 15:53
  • Leave it up; we'll just mark yours as a duplicate. As you said, searching for the right post is hard, and having these duplicates around makes things easier. – Ben Grossmann Apr 17 '18 at 15:57

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