Equivalence with Rayleigh Quotient Minimization

Golub (1973) showed that the constrained minimization problem of equations (1) and (2) is equivalent to minimization of the so-called Rayleigh Quotient. If we define the vector $\bold q = [\bold m \;\; -1]^T$ and $\bf A = [L \;\; d]$, the Rayleigh Quotient takes the following form:  

$$\mbox{min} \;\; F(\bold q) = \left\vert \frac{\bold q \bold A^T \bold A \bold q}{\bold q^T \bold q} \right\vert _2.$$ (3)

After the minimization of equation (3), the resultant vector q is the eigenvector associated with the smallest eigenvalue of $\bold A^T \bold A$.