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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Winter 2004 • Question 1

Question

Math 6610 • Analysis of Numerical Methods I Winter 2004 • Question 1

Householder Reflections A Householder Reflection \(H\) is an \(n\times n\) orthogonal matrix of the form \[ H=I-2uu^T \tag {1} \] where \[ \lVert u\rVert _2=1 \tag {2} \] and \(I\) is the \(n\times n\) identity matrix. Let \(a\in \mathbb {R}^n\) be a given (non-zero) vector. Show how to pick \(u\) in (1) such that \[ Ha=\lVert a\rVert _2 e_1 \tag {3} \] where \(e_1\) is the first unit vector. Describe how to use this technique to compute the \(QR\) factorization of an \(n\times n\) matrix \(A\).

Raw LaTeX

{Householder Reflections}
A Householder Reflection \(H\) is an \(n\times n\) orthogonal matrix of the form
\[
  H=I-2uu^T \tag{1}
\]
where
\[
  \lVert u\rVert_2=1 \tag{2}
\]
and \(I\) is the \(n\times n\) identity matrix. Let \(a\in\mathbb{R}^n\) be a given
(non-zero) vector. Show how to pick \(u\) in (1) such that
\[
  Ha=\lVert a\rVert_2 e_1 \tag{3}
\]
where \(e_1\) is the first unit vector. Describe how to use this technique to compute
the \(QR\) factorization of an \(n\times n\) matrix \(A\).

Topic Tags

Cholesky Factorization and SPD Matrices Linear Programming and Optimization Orthogonal and Unitary Matrices QR Factorization