Editing Rayleigh quotient (section)

==Bounds for Hermitian ''M''==
As stated in the introduction, for any vector ''x'', one has <math>R(M,x) \in \left[\lambda_\min, \lambda_\max \right]</math>, where <math>\lambda_\min, \lambda_\max</math> are respectively the smallest and largest eigenvalues of <math>M</math>. This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of ''M'':
<math display="block">R(M,x) = {x^{*} M x \over x^{*} x} = \frac{\sum_{i=1}^n \lambda_i y_i^2}{\sum_{i=1}^n y_i^2}</math>
where <math>(\lambda_i, v_i)</math> is the <math>i</math>-th eigenpair after orthonormalization and <math>y_i = v_i^* x</math> is the <math>i</math>th coordinate of ''x'' in the eigenbasis. It is then easy to verify that the bounds are attained at the corresponding eigenvectors <math>v_\min, v_\max</math>.

The fact that the quotient is a weighted average of the eigenvalues can be used to identify the second, the third, ... largest eigenvalues. Let <math>\lambda_{\max} = \lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n = \lambda_{\min} </math> 
be the eigenvalues in decreasing order. If <math>n=2</math> and <math>x</math> is constrained to be orthogonal to <math>v_1</math>, in which case <math>y_1 = v_1^*x = 0 </math>, then <math>R(M,x)</math> has maximum value <math>\lambda_2</math>, which is achieved when <math>x = v_2</math>.