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Linear Algebra With Applications
Found in: Page 224
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.


The QR factorization of the matrix is 212115=132-1/22-1/214/230323.

See the step by step solution

Step by Step Solution

Step 1: Determine column u→1 and entries r11 of R

Consider the matrix M=212115 where v1=221 and v2=115

By the theorem of QR method, the value of u1 and r11 is defined as follows:


Simplify the equation r11=||v1|| as follows:


Substitute the values 7 for r11 and 221 for v1 in the equation u1=1r11v1 as follows:


Therefore, the values localid="1659437134701" u1=2/32/31/3and r11=3 .

Step 2: Determine column  v→2⊥ and entries r12 of R

As r12=u1·v2, substitute the values 115 for V2 and 232313T for u1 in the equation

localid="1659437594586" r11=u1.V2 as follows:




Substitute the values 115 forV2 ,3 for r12 and 2/32/31/3 for u1 in the equation v2=V2-r12u1as follows:


v2=115-32/22/31/3v2=115-221v2=-1-14Therefore, v2=-1-14 and r12=3

Step 3: Determine column u→2  and entries r22 of R

The value of u2 and r22 is defined as follows.


Simplify the equation localid="1659438712674" r22=V2as follows.




Substitute the values 32 for r22 and -1-14 for V2 in the equation u2=1r22V2 as follows.


The values localid="1659439047262" u2=-1/32-1/324/32 and r22=32

Therefore, the matrices Q=2/3-1/322/3-1/321/34/32 and R=33032.

Hence, the QR factorization of the matrix is localid="1659439350112" 212115=132-1/22-1/214/233032.

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