8.4E: QR-Factorization Exercises
- Page ID
- 132840
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solutions
2
In each case find the QR-factorization of \(A\).
\(A = \left[ \begin{array}{rr} 1 & -1 \\ -1 & 0 \end{array}\right]\) \(A = \left[ \begin{array}{rr} 2 & 1 \\ 1 &1 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} 1 & 1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & -1 & 0 \end{array}\right]\)
- \(Q = \frac{1}{\sqrt{5}}\left[ \begin{array}{rr} 2 & -1 \\ 1 & 2 \end{array}\right]\), \(R = \frac{1}{\sqrt{5}}\left[ \begin{array}{rr} 5 & 3 \\ 0 & 1 \end{array}\right]\)
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\(Q = \frac{1}{\sqrt{3}}\left[ \begin{array}{rrr} 1 & 1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & -1 & 1 \end{array}\right]\),
\(R = \frac{1}{\sqrt{3}}\left[ \begin{array}{rrr} 3 & 0 & -1 \\ 0 & 3 & 1 \\ 0 & 0 & 2 \end{array}\right]\)
Let \(A\) and \(B\) denote matrices.
- If \(A\) and \(B\) have independent columns, show that \(AB\) has independent columns. [Hint: Theorem [thm:015672].]
- Show that \(A\) has a QR-factorization if and only if \(A\) has independent columns.
- [Hint: Consider \(AA^{T}\) where \(A = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 1 & 1 & 1 \end{array}\right]\).]
If \(A\) has a QR-factorization, use (a). For the converse use Theorem [thm:025133].
If \(R\) is upper triangular and invertible, show that there exists a diagonal matrix \(D\) with diagonal entries \(\pm 1\) such that \(R_{1} = DR\) is invertible, upper triangular, and has positive diagonal entries.
If \(A\) has independent columns, let
\(A = QR\) where \(Q\) has orthonormal columns and \(R\) is invertible and upper triangular. [Some authors call this a QR-factorization of \(A\).] Show that there is a diagonal matrix \(D\) with diagonal entries \(\pm 1\) such that \(A = (QD)(DR)\) is the QR-factorization of \(A\). [Hint: Preceding exercise.]