= Right singular vector
{title2=$v_j$}
A <right singular vector> $v_j$ is a <unit vector> in the input space satisfying $A^*Av_j=\sigma_j^2v_j$. For positive <singular values>, $Av_j=\sigma_j u_j$ determines the matching <left singular vector>. In the <ellipse> example, $Av_1=(5,0)^T$ is not parallel to $v_1$: a <right singular vector> need not be an <eigenvector> of $A$. Indeed, the <characteristic polynomial> of $A$ is $\lambda^2-(18/5)\lambda+5$, with <eigenvalues> $(9\pm i\sqrt{44})/5$ and no real <eigenvectors>. Nevertheless, its real <right singular vectors> give a complete <orthonormal basis>. <Eigenvectors> describe preserved directions; <right singular vectors> describe orthogonal input channels of stretching.https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/d273f75ee2552a5c3c35ccab37e5edce_MIT18_06SCF11_Ses3.5sum.pdf[MIT 18.06SC, Singular value decomposition: paired bases]{ref}
An arbitrary input mixes these channels. For $x=c_1v_1+c_2v_2$, <orthogonality> gives $\|Ax\|_2^2=25|c_1|^2+|c_2|^2$, while $\|x\|_2^2=|c_1|^2+|c_2|^2$. Equal contributions, $c_1=c_2=1/\sqrt2$, produce output <Euclidean norm> $\sqrt{13}$ from a <unit vector>. Concentrating the entire input along $v_1$ produces the maximum $5$. This is why the <right singular vectors> give a direct reading of how any input will behave. For a <compact operator>, the same definition describes the positive terms of its <singular value system>.
Back to article page
