Linear Algebra: UNIT IV: Matrix Decomposition
QR Decomposition, Singular Value Decomposition, Least Square Solutions: Exercise Problems
Matrix Decomposition
Linear Algebra: UNIT IV: Matrix Decomposition: QR Decomposition, Singular Value Decomposition, Least Square Solutions: Exercise Problems
QR Decomposition, Singular Value Decomposition, Least Square
Solutions
EXERCISE
1. Find the QR decomposition of the matrix
2.
Construct QR decomposition for the following matrix.
3. Construct the QR decomposition for the matrix
4. Construct the QR decomposition for the matrix
5.
Construct the QR decomposition for the matrix,
6.
Construct the QR decomposition of the matrix
7. Construct the QR decomposition of the matrix
8. Construct the QR decomposition for the matrix
9. Construct the QR decomposition for the matrix
10.
Use the shifted QR algorithm to determine all eigenvalues for the following
matrices.
11. Find the singular value decomposition for the matrix
12. Find the singular value decomposition for the matrix
13. Find the singular value decomposition for the matrix
14. Find the singular value decomposition for the matrix
15.
Find the singular value decomposition for the matrix
16. Find the singular value decomposition for the matrix
17.
Find the singular value decomposition for the matrix
18.
Find the least squares solutions to the given system of equations.
x1+x2+x3 = 1; x1 + x2+x3=2; x1 + x2+x3=3
[Ans: x1
= x2=x3 = 2/3 ⋅
19.
Find the least squares solutions of the form Ax=b where
20.
Find the least squares solutions for the system Ax=b where
[Ans: The system of equations are as x1+x3=5 and x2
‒ x3 = −3.
x1=5−x3
and x2=−3+x3
Let
x3=t, where t is any real number.
The general least squares solution is given by 
This
represents an infinite set of solutions.]
21.
Find the least squares solution to the given system of equations.
(a)
2x+3y=8
3x−y=5
x+y=6
(b) 2x+y=8
y=4
−x+y=0
3x+y=13
(c)
x+3y=65
2x
− y = 0
3x+y=50
2x+2y=55
(d)
2x+y=6A
x+y=8
−
2x + y = 11
−x+y=8
3x+y=4
(e)
2x+3y−4z=1
x−2y+3z=3
x+4y+2z=6
2x+y−3z=1
(f) 2x+3y+2z = 25
2x−y+3z=30
3x+4y−2z=20
3x+5y+4z=
55
(g)
x+2y+2z=1
2x+3y+2z=2
2x+4y+4z=−2
3x+5y+4z=1
x+3y+2z=−1
22.
Find the least squares straight line for the given xy data:
[Ans: y = (2.3)x+8.1]
23.
Find the least squares straight line that best fits this xy data.
[Ans: y=(−2.6)x+54.4]
24.
Find the least squares straight line that best fits this data.
[Ans: y = (0.27) x + 10.24]
Linear Algebra: UNIT IV: Matrix Decomposition : Tag: maths, mathematics : Matrix Decomposition - QR Decomposition, Singular Value Decomposition, Least Square Solutions: Exercise Problems
Linear Algebra: UNIT IV: Matrix Decomposition
Under Subject
Linear Algebra
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