 Left singular vector

A left singular vector $u_j$ is a unit vector in the output space satisfying $A{A}^{\ast }{u}_j={\sigma }_j^2{u}_j$. For positive singular values, the pairing also satisfies $A^{\ast }{u}_j={\sigma }_j{v}_j$. 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 $A^{\ast }$. 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 $A{v}_j={\sigma }_j{u}_j$ and the outer product $u_j{v}_j^{\ast }$. Reversing only one arrow would change that term's sign. The paired directions, rather than independently chosen arrow orientations, encode the matrix.