Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Let T: VV be a linear operator on a finite-dimensional F-vector space V, and let f(t) be the characteristic polynomial of T. Show that T is diagonalizable if and only if f(t) splits over F and rank(T - XI) = rank((T – XI)²) for every eigenvalue A of T. (You can use the theorem that T has a Jordan basis iff f(t) splits, and the theorem that says nullity ((T – XI)²) – nullity (T – XI) = ···)

Let T: VV be a linear operator on a finite-dimensional F-vector space V, and let f(t) be the characteristic polynomial of T. Show that T is diagonalizable if and only if f(t) splits over F and rank(T - XI) = rank((T – XI)²) for every eigenvalue A of T. (You can use the theorem that T has a Jordan basis iff f(t) splits, and the theorem that says nullity ((T – XI)²) – nullity (T – XI) = ···)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
100%
5. Let T: V→ V be a linear operator on a finite-dimensional F-vector space V, and let f(t) be the characteristic polynomial of T. Show that T is diagonalizable if and only if f(t) splits over F and rank(T – XI) = rank((T – XI)²) for every eigenvalue A of T. (You can use the theorem that T has a Jordan basis iff f(t) splits, and the theorem that says nullity ((T – XI)²) – nullity(T – XI) =···)
Transcribed Image Text:5. Let T: V→ V be a linear operator on a finite-dimensional F-vector space V, and let f(t) be the characteristic polynomial of T. Show that T is diagonalizable if and only if f(t) splits over F and rank(T – XI) = rank((T – XI)²) for every eigenvalue A of T. (You can use the theorem that T has a Jordan basis iff f(t) splits, and the theorem that says nullity ((T – XI)²) – nullity(T – XI) =···)
Expert Solution
steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS