Left singular vector ($u_j$)

A unit left singular vector of a matrix or compact operator $A$ is an eigenvector of $A{A}^{\ast }$ with eigenvalue ${\sigma }_j^2$. For ${\sigma }_j>0$, it pairs with a right singular vector $v_j$ through $A{v}_j={\sigma }_j{u}_j$ and $A^{\ast }{u}_j={\sigma }_j{v}_j$.