As we know that the famous Rayleigh quotient is definite as $\frac{x^TAx}{x^Tx}$

The maximum value is equals to the maximum eigenvalue of matrix $A$ when it is symmetric

$\max_{x^Tx=1} \frac{x^TAx}{x^Tx} = \lambda_{max}(A)$

Yet how about the search space is being further limited?

For example, in this case how will the optimal value of Rayleigh quotient behave?

$\max_{x^Tx=1,x\in S} \frac{x^TAx}{x^Tx}$

where $S = \{ x|x=b-Ma,M\in \mathbb{D}\}$, $\mathbb{D}$ is the set of all doubly-stochastic matrices

  • $\begingroup$ If $A$ is not real symmetric, it is not necessarily true that $\max_{x^Tx=1} \frac{x^TAx}{x^Tx} = \lambda_{max}(A)$. $\endgroup$ – user1551 Apr 11 '13 at 10:41
  • $\begingroup$ @user1551 Sorry updated. $\endgroup$ – Rein Apr 12 '13 at 3:41

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