THEOREM 2.3. Let X be a finite-dimensional inner product space and let E be a projection. Then the following statements are equivalent. (1) E is normal; (2) E is self-adjoint; (3) E is the orthogonal projection on its range. Proof (3) ⇒ (1). Let x, y e X. Then, since E is idempotent, x - Ex = N(E) = R(E)¹ Ey = R(E). (x - Ex, Ey)=0 and Thus or (x, Ey) = (Ex, Ey) = (x, E*Ey) (for any x, y € X). Therefore E = E*E and the proof becomes the same as in proving (1) ⇒ (2); that is, it follows that E is self-adjoint, which certainly implies that E is normal, and com- pletes the proof of the theorem.
THEOREM 2.3. Let X be a finite-dimensional inner product space and let E be a projection. Then the following statements are equivalent. (1) E is normal; (2) E is self-adjoint; (3) E is the orthogonal projection on its range. Proof (3) ⇒ (1). Let x, y e X. Then, since E is idempotent, x - Ex = N(E) = R(E)¹ Ey = R(E). (x - Ex, Ey)=0 and Thus or (x, Ey) = (Ex, Ey) = (x, E*Ey) (for any x, y € X). Therefore E = E*E and the proof becomes the same as in proving (1) ⇒ (2); that is, it follows that E is self-adjoint, which certainly implies that E is normal, and com- pletes the proof of the theorem.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Request explain the highlighted portion
Transcribed Image Text:THEOREM 2.3. Let X be a finite-dimensional inner product space and let E be a
projection. Then the following statements are equivalent.
(1) E is normal;
(2) E is self-adjoint;
(3) E is the orthogonal projection on its range.
Proof (3) ⇒ (1). Let x, y = X. Then, since E is idempotent,
x - Ex = N(E) = R(E)¹
Ey = R(E).
(x - Ex, Ey) = 0
and
Thus
or
(x, Ey) = (Ex, Ey) = (x, E* Ey)
(for any x, y e X).
Therefore E = E*E and the proof becomes the same as in proving (1) ⇒ (2); that is,
it follows that E is self-adjoint, which certainly implies that E is normal, and com-
pletes the proof of the theorem.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
