a. A", n21 b. kA, k any scalar. c. p(A), p(x) any polynomial (Theorem 3.3.1) b. Show that 2³-22+3 is an eigenvalue of A³-2A +31. c. Show that p(2) is an eigenvalue of p(A) for any nonzero polynomial p(x). Exercise 3.3.21 Suppose λ is an eigenvalue of a square matrix A with eigenvector x 0. a. Show that 22 is an eigenvalue of A² (with the same x). 3.3. Diagonalization and Eigenvalues 189 Exercise 3.3.25 Let A² = I, and assume that A# I and A#-I. a. Show that the only eigenvalues of A are λ = 1 and λ = -1. b. Show that A is diagonalizable. [Hint: Verify that A(A+1)=A+I and A(A-1)=-(A-I), and then look at nonzero columns of A+I and of A-Z.]
a. A", n21 b. kA, k any scalar. c. p(A), p(x) any polynomial (Theorem 3.3.1) b. Show that 2³-22+3 is an eigenvalue of A³-2A +31. c. Show that p(2) is an eigenvalue of p(A) for any nonzero polynomial p(x). Exercise 3.3.21 Suppose λ is an eigenvalue of a square matrix A with eigenvector x 0. a. Show that 22 is an eigenvalue of A² (with the same x). 3.3. Diagonalization and Eigenvalues 189 Exercise 3.3.25 Let A² = I, and assume that A# I and A#-I. a. Show that the only eigenvalues of A are λ = 1 and λ = -1. b. Show that A is diagonalizable. [Hint: Verify that A(A+1)=A+I and A(A-1)=-(A-I), and then look at nonzero columns of A+I and of A-Z.]
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
Please answer Exercise 3.3.21 c).
![1
b. If D =
find a diagonalizable matrix A Exercise 3.3.20 Let A be an invertible nxn matrix.
such that D+A is not diagonalizable.
0-1
0
Exercise 3.3.10 If A is an nx n matrix, show that A is
diagonalizable if and only if AT is diagonalizable.
Exercise 3.3.11 If A is diagonalizable, show that each
of the following is also diagonalizable.
a. A", n ≥ 1
b. kA, k any scalar.
c. p(A), p(x) any polynomial (Theorem 3.3.1)
b. Show that 23-22+3 is an eigenvalue of
A³-24 +31.
c. Show that p(2) is an eigenvalue of p(A) for any
nonzero polynomial p(x).
Exercise 3.3.22 If A is an n x n matrix, show that
CA2 (x²)=(-1)"CA(x)CA(-x).
Exercise 3.3.23 An n x n matrix A is called nilpotent if
Am=0 for some m > 1.
a. Show that every triangular matrix with zeros on
the main diagonal is nilpotent.
b. If A is nilpotent, show that λ = 0 is the only eigen-
value (even complex) of A.
a. Show that the eigenvalues of A are nonzero.
b. Show that the eigenvalues of A are precisely the
numbers 1/2, where is an eigenvalue of A.
c. Show that CA-1(x) = (CA (+).
det
Exercise 3.3.21 Suppose λ is an eigenvalue of a square
matrix A with eigenvector x = 0.
a. Show that 22 is an eigenvalue of A² (with the same
x).
3.3. Diagonalization and Eigenvalues 189
Exercise 3.3.25 Let A² = I, and assume that A # I and
A#-I.
a. Show that the only eigenvalues of A are λ = 1 and
λ = -1.
b. Show that A is diagonalizable. [Hint: Verify that
A(A+1)=A+I and A (A-1)=-(A-I), and then
look at nonzero columns of A+I and of A-1.]
c. If Qm: R² R2 is reflection in the line y = mx
where m0, use (b) to show that the matrix of
Qm is diagonalizable for each m.
d. Now prove (c) geometrically using Theorem 3.3.3.
Exercise 3.3.26 Let A =
010
301 . Show that CA (x) = CB(x) = (x + 1)²(x-
200
2), but A is diagonalizable and B is not.
Exercise 3.3.27
23-3
1 0 -1 and B =
11 -2
a. Show that the only diagonalizable matrix A that
has only one eigenvalue is the scalar matrix
A = 21.
1c [3 -2
h Is
diagonalizable?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca54911f-4c33-48b6-9518-41b412a01cf9%2F2c9ab3f8-1847-4eaf-8c0a-a686ee8012e0%2Fbkufrf4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1
b. If D =
find a diagonalizable matrix A Exercise 3.3.20 Let A be an invertible nxn matrix.
such that D+A is not diagonalizable.
0-1
0
Exercise 3.3.10 If A is an nx n matrix, show that A is
diagonalizable if and only if AT is diagonalizable.
Exercise 3.3.11 If A is diagonalizable, show that each
of the following is also diagonalizable.
a. A", n ≥ 1
b. kA, k any scalar.
c. p(A), p(x) any polynomial (Theorem 3.3.1)
b. Show that 23-22+3 is an eigenvalue of
A³-24 +31.
c. Show that p(2) is an eigenvalue of p(A) for any
nonzero polynomial p(x).
Exercise 3.3.22 If A is an n x n matrix, show that
CA2 (x²)=(-1)"CA(x)CA(-x).
Exercise 3.3.23 An n x n matrix A is called nilpotent if
Am=0 for some m > 1.
a. Show that every triangular matrix with zeros on
the main diagonal is nilpotent.
b. If A is nilpotent, show that λ = 0 is the only eigen-
value (even complex) of A.
a. Show that the eigenvalues of A are nonzero.
b. Show that the eigenvalues of A are precisely the
numbers 1/2, where is an eigenvalue of A.
c. Show that CA-1(x) = (CA (+).
det
Exercise 3.3.21 Suppose λ is an eigenvalue of a square
matrix A with eigenvector x = 0.
a. Show that 22 is an eigenvalue of A² (with the same
x).
3.3. Diagonalization and Eigenvalues 189
Exercise 3.3.25 Let A² = I, and assume that A # I and
A#-I.
a. Show that the only eigenvalues of A are λ = 1 and
λ = -1.
b. Show that A is diagonalizable. [Hint: Verify that
A(A+1)=A+I and A (A-1)=-(A-I), and then
look at nonzero columns of A+I and of A-1.]
c. If Qm: R² R2 is reflection in the line y = mx
where m0, use (b) to show that the matrix of
Qm is diagonalizable for each m.
d. Now prove (c) geometrically using Theorem 3.3.3.
Exercise 3.3.26 Let A =
010
301 . Show that CA (x) = CB(x) = (x + 1)²(x-
200
2), but A is diagonalizable and B is not.
Exercise 3.3.27
23-3
1 0 -1 and B =
11 -2
a. Show that the only diagonalizable matrix A that
has only one eigenvalue is the scalar matrix
A = 21.
1c [3 -2
h Is
diagonalizable?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
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,
