Chapter 4.6 The QR Algorithm

Schur's Thm

$\exists Q$ unitary $\rightarrow Q^* A Q = T = D + N$

If A is given , and u₀ is any unitary matrix

A₁ = u₀^* A₀ u₀

$\left \{ \matrix{ A_1 = Q_2 R_2 \cr set ~~R_2 Q_2 = A_2 \cr} \right$ $\left \matrix{ {\left \{ \matrix{ A_0 = Q_1 R_1 \cr R_1 Q_1 = A_1 \cr } ... ... \{ \matrix{ A_1 = Q_2 R_2 \cr R_2 Q_2 = A_2 \cr } \right } \cr } \right$ general form $\left \{ \matrix{ A_{m-1} = Q_m R_m \cr R_m Q_m = A_m \cr } \right$

[Thm] $\{ A_m \} \overrightarrow { m \rightarrow \infty } T = D + N$ Note : $A_m = {Q_m}^* A_{m-1} Q_m \Rightarrow A_m = ( {Q_m}^* {Q_{m-a}}^* \cdots {Q_1}^* ) A_0 ( Q_1 \cdots Q_m )$

$\Vert ~~~~~~~~~\Vert \par R_m \cdot Q_m ~~~~~{Q_m}^* (Q_m \cdot R_m ) Q_m$

$A = \left [ \matrix{ 8 & 2 \cr 2 & 5 \cr } \right ] \par A = A_0 = Q_1 R_1$

Let $A_1 = R_1 Q_1 = \left [ \matrix{ 8.7648 & 1.0588 \cr 1.0588 & 4.2352 \cr } \right ]$

Upper Hessenberg

$\left [ \matrix{ * & * & * & \cdots & \cdots & * \cr * & * & * &&& \vdots \cr... ... && \vdots \cr &&& \ddots & \ddots & \vdots \cr 0 &&&& * & * \cr } \right ]$