8.4E: QR-Factorization Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercises for 1
solutions
2
In each case find the QR-factorization of A.
A=[1−1−10] A=[2111] A=[111110100000] A=[110−1010111−10]
- Q=1√5[2−112], R=1√5[5301]
-
Q=1√3[110−1010111−11],
R=1√3[30−1031002]
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 AAT where A=[100111].]
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 ±1 such that R1=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 ±1 such that A=(QD)(DR) is the QR-factorization of A. [Hint: Preceding exercise.]