(a) Show that A is an n x n matrix and det(A) # 0, then the projection matrix P A(ATA)-¹A¹ is the nxn identity matrix I (matrix with 1 on the diagonal and 0 outside of the diagonal).
(a) Show that A is an n x n matrix and det(A) # 0, then the projection matrix P A(ATA)-¹A¹ is the nxn identity matrix I (matrix with 1 on the diagonal and 0 outside of the diagonal).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Projection Matrix
(a) Show that A is an n x n matrix and det(A) 0, then the projection matrix P
A(ATA)-¹AT is the nxn identity matrix I (matrix with 1 on the diagonal and 0 outside
of the diagonal).
(b) Consider the system equations r = 4 and r = 6. Obviously this system of equations
does not have an exact solution. Represent the system of equations as Av = b where
A = [1,1], v = [r], and b = [4,6]T. Find the projection matrix P. Show graphically
the point b (the first coordinate on the z-axis and the second coordinate on the y-axis)
and the point Pb. Show the set of points v € R2x1 such that Pu= v. What is the value
of a that solves Az = Pb?
(c) Repeat the previous exercise for the system of equations r = 4 and 2x = 12. Keep in
mind that you will obtain a different projection matrix. What is the value of a that
solves Ax = Pb?
(d) Find the approximate solution of the system of equations (using the projection matrix)
[1 2]
34
5 6
] [C]
10
15](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fa6da54-4e28-4e8d-979d-89b29b7ed7e0%2Fc50ec904-414f-415a-ab86-76ecdb645a10%2Fx2f9lso_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Projection Matrix
(a) Show that A is an n x n matrix and det(A) 0, then the projection matrix P
A(ATA)-¹AT is the nxn identity matrix I (matrix with 1 on the diagonal and 0 outside
of the diagonal).
(b) Consider the system equations r = 4 and r = 6. Obviously this system of equations
does not have an exact solution. Represent the system of equations as Av = b where
A = [1,1], v = [r], and b = [4,6]T. Find the projection matrix P. Show graphically
the point b (the first coordinate on the z-axis and the second coordinate on the y-axis)
and the point Pb. Show the set of points v € R2x1 such that Pu= v. What is the value
of a that solves Az = Pb?
(c) Repeat the previous exercise for the system of equations r = 4 and 2x = 12. Keep in
mind that you will obtain a different projection matrix. What is the value of a that
solves Ax = Pb?
(d) Find the approximate solution of the system of equations (using the projection matrix)
[1 2]
34
5 6
] [C]
10
15
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