Exercise30.1
Use that the square matrix \(A\) has a \(QR\) factorisation to prove that
\begin{equation*} |\det(A)| \leq \prod_{j=1}^m \|a_j\|_2, \end{equation*}where, as usual, \(a_j\) is column \(j\) of \(A\text{.}\)
Use that the square matrix \(A\) has a \(QR\) factorisation to prove that
\begin{equation*} |\det(A)| \leq \prod_{j=1}^m \|a_j\|_2, \end{equation*}where, as usual, \(a_j\) is column \(j\) of \(A\text{.}\)
Let \(F\) be a Householder reflector. That is, for some vector \(v\text{,}\)
\begin{equation*} F = I - 2 \frac{v v^\star}{v^\star v}. \end{equation*}Determine the eigenvalues, determinant, and singular values of \(F\text{.}\)