 Right singular vector

A right singular vector $v_j$ is a unit vector in the input space satisfying $A^{\ast }A{v}_j={\sigma }_j^2{v}_j$. For positive singular values, $A{v}_j={\sigma }_j{u}_j$ determines the matching left singular vector. In the ellipse example, $A{v}_1=(5,0{)}^T$ 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.^[ref]

An arbitrary input mixes these channels. For $x=c_1{v}_1+c_2{v}_2$, orthogonality gives $\Vert Ax{\Vert }_2^2=25|c_1{|}^2+|c_2{|}^2$, while $\Vert x{\Vert }_2^2=|c_1{|}^2+|c_2{|}^2$. Equal contributions, $c_1=c_2=1/\sqrt{2}$, 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.