Page 359 - Matrix Analysis & Applied Linear Algebra

5.7 Orthogonal Reduction 355

m×n
5.7.2. For A ∈ , suppose that rank (A)= n, and let P be an orthog-
onal matrix such that

R n×n
PA = T = ,
0
where R is an upper-triangular matrix. If P T is partitioned as
T
P =[X m×n | Y] ,
explain why the columns of X constitute an orthonormal basis for
R (A).

5.7.3. By using Householder reduction, find an orthonormal basis for R (A),
where
 
4 −3 4
 2 −14 −3 
−2 14 0
A =   .
1 −7 15
5.7.4. Use Householder reduction to compute the least squares solution for
Ax = b, where
   
4 −3 4 5
 2 −14 −3   −15 
−2 14 0 0
A =   and b =   .
1 −7 15 30
Hint: Make use of the factors you computed in Exercise 5.7.3.

5.7.5. If A = QR is the QR factorization for A, explain why A = R ,
F F
where is the Frobenius matrix norm introduced on p. 279.
F
T
5.7.6. Find an orthogonal matrix P such that P AP = H is in upper-
Hessenberg form, where
 
−2 3 −4
A =  3 −25 50  .
−4 50 25

5.7.7. Let H be an upper-Hessenberg matrix, and suppose that H = QR,
where R is a nonsingular upper-triangular matrix. Prove that Q as
well as the product RQ must also be in upper-Hessenberg form.

5.7.8. Approximately how many multiplications are needed to reduce an n × n
nonsingular upper-Hessenberg matrix to upper-triangular form by using
plane rotations?