4. a) b) i) i., iii, combinations of I and H. i) (- Let H = 5 7 Find the characteristic polynomial of H. 'Is H diagonalisable? Explain your answer. Use the Cayley-Hamilton theorem to write H-¹ and H³ as linear Show that the matrix A=(12) has only one eigenvalue and that is the corresponding eigenvector. ii)] Solve the system of differential equations U r' (t)=-4y+ 3et y' (t)=9r+12y-4e6t; V = with initial conditions r(0) = 2, y(0) = -3. Let F = FX-XF. You may assume without proof that the map f is linear. i) [ -- Prove that ƒ²(X) = −2FXF, i.e., prove that ƒ(ƒ(X)) = −2F XF. ….. Prove that ƒ³ (X) = 0, for any X € M₂,2(R), i.e., prove that ƒ (ƒ(ƒ(X))) = 0, for any X € M2,2(R). Let E11 C · († 8). Define the map f: M22(R) → M2,2(R) by f(X) = = (9). E₁2 - ( ). En - (1₂9). En-( 1). = be the standard basis in M₂,2 (R). A) Find f(E11), f(E12), f(E21), f(E22). B) Find the matrix, M, of the map f with respect to the standard basis in the domain and in the co-domain. C) Without computing, explain why M² #0 and M³ = 0. D) Let J be the Jordan Normal form of the matrix M, i.e., M~ J. Explain why 0
4. a) b) i) i., iii, combinations of I and H. i) (- Let H = 5 7 Find the characteristic polynomial of H. 'Is H diagonalisable? Explain your answer. Use the Cayley-Hamilton theorem to write H-¹ and H³ as linear Show that the matrix A=(12) has only one eigenvalue and that is the corresponding eigenvector. ii)] Solve the system of differential equations U r' (t)=-4y+ 3et y' (t)=9r+12y-4e6t; V = with initial conditions r(0) = 2, y(0) = -3. Let F = FX-XF. You may assume without proof that the map f is linear. i) [ -- Prove that ƒ²(X) = −2FXF, i.e., prove that ƒ(ƒ(X)) = −2F XF. ….. Prove that ƒ³ (X) = 0, for any X € M₂,2(R), i.e., prove that ƒ (ƒ(ƒ(X))) = 0, for any X € M2,2(R). Let E11 C · († 8). Define the map f: M22(R) → M2,2(R) by f(X) = = (9). E₁2 - ( ). En - (1₂9). En-( 1). = be the standard basis in M₂,2 (R). A) Find f(E11), f(E12), f(E21), f(E22). B) Find the matrix, M, of the map f with respect to the standard basis in the domain and in the co-domain. C) Without computing, explain why M² #0 and M³ = 0. D) Let J be the Jordan Normal form of the matrix M, i.e., M~ J. Explain why 0
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.5: Iterative Methods For Computing Eigenvalues
Problem 42EQ
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Question
Part C(III) needed
Needed to be solved Part C(III) Correctly in 15 minutes and get the thumbs up please show neat and clean work for it
![4. a)
i)
i.,
iii,
L
-9
Let H = (-
= (-3--²³)
5
combinations of I and H.
V=
Find the characteristic polynomial of H.
Is H diagonalisable? Explain your answer.
Use the Cayley-Hamilton theorem to write H-1 and H³ as linear
b)
i) [---] Show that the matrix A = (12) has only one eigenvalue and that
is the corresponding eigenvector.
ii)] Solve the system of differential equations
with initial conditions (0) = 2, y(0) = -3.
r' (t)=-4y+ 3et
y' (t) = 9x + 12y-4e6t;
U
PRO
Let F =
FX-XF. You may assume without proof that the map f is linear.
i) [
-- Prove that f²(X) = -2FXF, i.e., prove that f(f(x)) = −2FXF.
Prove that ƒ³ (X) = 0, for any X € M2,2(R), i.e., prove that f(f(f(x))) =
0, for any Xe M2,2 (R).
Let
- (8) Define the map f: M2,2(R) → M2,2(R) by f(X) =
be the standard basis in M2,2 (R).
A) Find
1
=
E11
- (₁₂9). ₂- (₂2). ₂ - (2), E₂-(2).
E12 =
E21 =
f(E1), f(E12), f(E21), f(E22).
B) Find the matrix, M, of the map f with respect to the standard basis in the
domain and in the co-domain.
C) Without computing, explain why M² #0 and M³ = 0.
D)
Let J be the Jordan Normal form of the matrix M, i.e., MJ. Explain why 0
is the only eigenvalue of the matrix J (and of M).
E)
Find the Jordan normal form of the matrix M.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2e672a1-54a8-4743-aad6-f3d1c4a0c4ab%2Fc1c3c3a2-395d-4066-8fd8-46c49836af52%2F4uklrs6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. a)
i)
i.,
iii,
L
-9
Let H = (-
= (-3--²³)
5
combinations of I and H.
V=
Find the characteristic polynomial of H.
Is H diagonalisable? Explain your answer.
Use the Cayley-Hamilton theorem to write H-1 and H³ as linear
b)
i) [---] Show that the matrix A = (12) has only one eigenvalue and that
is the corresponding eigenvector.
ii)] Solve the system of differential equations
with initial conditions (0) = 2, y(0) = -3.
r' (t)=-4y+ 3et
y' (t) = 9x + 12y-4e6t;
U
PRO
Let F =
FX-XF. You may assume without proof that the map f is linear.
i) [
-- Prove that f²(X) = -2FXF, i.e., prove that f(f(x)) = −2FXF.
Prove that ƒ³ (X) = 0, for any X € M2,2(R), i.e., prove that f(f(f(x))) =
0, for any Xe M2,2 (R).
Let
- (8) Define the map f: M2,2(R) → M2,2(R) by f(X) =
be the standard basis in M2,2 (R).
A) Find
1
=
E11
- (₁₂9). ₂- (₂2). ₂ - (2), E₂-(2).
E12 =
E21 =
f(E1), f(E12), f(E21), f(E22).
B) Find the matrix, M, of the map f with respect to the standard basis in the
domain and in the co-domain.
C) Without computing, explain why M² #0 and M³ = 0.
D)
Let J be the Jordan Normal form of the matrix M, i.e., MJ. Explain why 0
is the only eigenvalue of the matrix J (and of M).
E)
Find the Jordan normal form of the matrix M.
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