Right singular vector ($v_j$)

A unit right singular vector of a matrix or compact operator $A$ is an eigenvector of $A^{\ast }A$ with eigenvalue ${\sigma }_j^2$. For ${\sigma }_j>0$, $u_j=A{v}_j/{\sigma }_j$ is the corresponding left singular vector. The largest singular value is the operator norm of $A$, attained on its corresponding right singular vectors.