Skip to main content
\(\newcommand{\Cs}{\mathbb{Cs}} \newcommand{\Cm}{\mathbb{C}^{m}} \newcommand{\Cmm}{\mathbb{C}^{m\times m}} \newcommand{\Cmn}{\mathbb{C}^{m\times n}} \newcommand{\R}{\mathbb{R}} \newcommand{\Rm}{\mathbb{R}^{m}} \newcommand{\Rn}{\mathbb{R}^{n}} \newcommand{\Rmm}{\mathbb{R}^{m\times m}} \DeclareMathOperator{\Range}{range} \DeclareMathOperator{\Rank}{rank} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\Null}{null} \newcommand{\definiteintegral}[4]{\int_{#1}^{#2}\,#3\,d#4} \newcommand{\indefiniteintegral}[2]{\int#1\,d#2} \graphicspath{ {images/} } \usepackage{amsmath} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section34Hessenberg Form and the QR Algorithm. (NM, 21 Nov)

Subsection34.1Exercise

Recall that the \(QR\) Algorithm is:

  • Set \(A^{(0)} = A\)
  • \(k=1, 2, 3, \dots \)
    • Compute \(Q^{(k)}R^{(k)}=A\text{,}\) the QR factorisation of \(A\text{.}\)
    • Set \(A^{(k)} = Q^{(k)}R^{(k)}\)
Exercise34.1

Show that \(A^{(k)}\) is similar \(A\text{.}\)

Exercise34.2

Show that \(A^k = \underline{Q}^{(k)}\underline{R}^{(k)}\text{,}\) where

\begin{equation*} \underline{Q}^{(k)} = Q^{(1)}Q^{(2)} \cdots Q^{(k)}, \end{equation*}

and

\begin{equation*} \underline{R}^{(k)} = R^{(k)}R^{(k-1)} \cdots R^{(1)}. \end{equation*}