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Mathematics-Online course: Linear Algebra - Normal Forms - Diagonalisation

Rayleigh Quotient

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Let $ S$ be a symmetric positive definite matrix. Then the extreme values of quotient

$\displaystyle r_S(x) = \frac{x^{\operatorname t}S x}{x^{\operatorname t}x},\quad x\ne 0\, , $

coincide with the smallest and the largest eigenvalue of $ S$.
(Authors: App/Burkhardt/Höllig)
The matrix $ S$ can be transformed into diagonal form by a unitary matrix $ U$:

$\displaystyle U^{\operatorname t}S U = \Lambda,\quad\lambda_i>0\, . $

Substituting $ S = U\Lambda U^{\operatorname t}$ and $ x = Uy$ we obtain

$\displaystyle r_S(y) = \frac{y^{\operatorname t}\Lambda y}{y^{\operatorname t}... ...) y} =\frac{\sum_i \lambda_i \vert y_i\vert^2}{\sum_i \vert y_i\vert^2}\, . $

This proves that

$\displaystyle \lambda_{\min} \le r_S(y) \le \lambda_{\max}\, , $

and we have equality for the corresponding eigenvectors $ x$.
(Authors: App/Burkhardt/Höllig)
  automatically generated 4/21/2005