maximum and miniumum of this expression

Question

I am asked to find max and min of $L=\sum_{1\le i,j\le n} a_{ij}x_ix_j$ on the $n$ dimensional unit sphere.

I am simply not getting this how to proceed.

Thanks for helping.

Answer 1

This is equivalent to extremizing $\frac{x^TAx}{x^Tx}$, where $[A]_{ii}=2a_{ii}$ and $[A]_{ij}=a_{ij}$ when $i\neq j$. Also known as the Rayleigh quotient of a matrix, if $A$ is diagonalizable then the answer is the largest and smallest eigenvalues of $A$. Otherwise the max and min are equal to the largest and smallest singular values of $A$. Thus one way of solving this question is to compute the eigenvalues of $A^TA$ and then take the largest and smallest eigenvalues.