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Mathematics LibreTexts

8.4E: QR-Factorization Exercises

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Exercises for 1

solutions

2

In each case find the QR-factorization of A.

A=[1110] A=[2111] A=[111110100000] A=[110101011110]

  1. Q=15[2112], R=15[5301]
  2. Q=13[110101011111],
    R=13[301031002]

Let A and B denote matrices.

  1. If A and B have independent columns, show that AB has independent columns. [Hint: Theorem [thm:015672].]
  2. Show that A has a QR-factorization if and only if A has independent columns.
  3. [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.]


8.4E: QR-Factorization Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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