I'm reading this paper, which uses the quantity $$\max_{x\neq0} \frac{x^T A x}{x^Tx}$$ where $A\in R^{n\times n}$ is nonsingular and $x\in R^n$.
This quantity looks so familiar to me that I'm almost certain this quantity has a special name in linear algebra... Does anyone recognize it or know its name?

  • 1
    $\begingroup$ The quantity inside the $\max$ is known as a Rayleigh quotient when $A$ is symmetric (Hermitian). $\endgroup$ – copper.hat Feb 28 '13 at 20:37
  • $\begingroup$ Thanks @copper.hat! I presume that we could also limit our search for a maximum to the unit ball $||x||=1$, correct? $\endgroup$ – Paul Feb 28 '13 at 20:41
  • $\begingroup$ The boundary of the unit ball. (If $A = -I$, then the max would be attained at $x=0$ if you maximized over the unit ball.) $\endgroup$ – copper.hat Feb 28 '13 at 20:48

The entire term is the matrix norm of A and the objective function is the Rayleigh quotient.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.