Editing Rayleigh quotient (section)

=== Formulation using Lagrange multipliers ===
Alternatively, this result can be arrived at by the method of [[Lagrange multipliers]]. The first part is to show that the quotient is constant under scaling <math>x \to cx</math>, where <math>c</math> is a scalar
<math display="block">R(M,cx) = \frac {(cx)^{*} M cx} {(cx)^{*} cx} = \frac {c^{*} c} {c^{*} c} \frac {x^{*} M x} {x^{*} x} = R(M,x).</math>

Because of this invariance, it is sufficient to study the special case <math>\|x\|^2 = x^Tx = 1</math>. The problem is then to find the [[critical point (mathematics)|critical points]] of the function
<math display="block">R(M,x) = x^\mathsf{T} M x ,</math>
subject to the constraint <math>\|x\|^2 = x^Tx = 1.</math> In other words, it is to find the critical points of 
<math display="block">\mathcal{L}(x) = x^\mathsf{T} M x  -\lambda \left (x^\mathsf{T} x - 1 \right), </math>
where <math>\lambda</math> is a Lagrange multiplier. The stationary points of <math>\mathcal{L}(x)</math> occur at
<math display="block">\begin{align} 
&\frac{d\mathcal{L}(x)}{dx} = 0 \\
\Rightarrow{}& 2x^\mathsf{T}M  - 2\lambda x^\mathsf{T} = 0 \\
\Rightarrow{}& 2Mx  - 2\lambda x = 0 \text{ (taking the transpose of both sides and noting that }M\text{ is Hermitian)}\\
\Rightarrow{}& M x = \lambda x
\end{align} </math>
and 
<math display="block"> \therefore R(M,x) = \frac{x^\mathsf{T} M x}{x^\mathsf{T} x} = \lambda \frac{x^\mathsf{T}x}{x^\mathsf{T} x} = \lambda.</math>

Therefore, the eigenvectors <math>x_1, \ldots, x_n</math> of <math>M</math> are the critical points of the Rayleigh quotient and their corresponding eigenvalues <math>\lambda_1, \ldots, \lambda_n</math> are the stationary values of <math>\mathcal{L}</math>. This property is the basis for [[principal components analysis]] and [[canonical correlation]].