= Left singular vector
{title2=$u_j$}
A <left singular vector> $u_j$ is a <unit vector> in the output space satisfying $AA^*u_j=\sigma_j^2u_j$. For positive <singular values>, the pairing also satisfies $A^*u_j=\sigma_jv_j$. Thus $u_1=(1,0)^T$ and $u_2=(0,1)^T$ describe the output axes of the preceding <ellipse>, while the <right singular vectors> describe its input directions. At zero <singular values>, division by $\sigma_j$ is impossible: complete the <orthonormal bases> independently in the <null spaces> of $A$ and $A^*$. Repeated <singular values> allow different matched <orthonormal bases> within the corresponding <eigenspaces>. The same positive pairing extends to a <compact operator> through its <singular value system>.
Even distinct <singular values> do not fix arrow signs: replacing both $u_j$ and $v_j$ by their negatives preserves $Av_j=\sigma_j u_j$ and the <outer product> $u_jv_j^*$. Reversing only one arrow would change that term's sign. The paired directions, rather than independently chosen arrow orientations, encode the <matrix>.
Back to article page
