1. Let A € M4(F) and let J € M₂(F) be a Jordan canonical form of A (you may assume A = J to do the questions below). Find J and the characteristic and minimal polynomials pÃ(™), m₁(x) of A for each case below, where A₁,..., Ak are the only distinct eigenvalues of A. Hint. See T8.7 and its General remark. k 1a. Σ₁ dim V₁, (A) = 4. [Answer: J is diagonal (5 cases).] i=1 1b. 1c. 1 dim Vx, (A) = 3 and k = 3. [Answer: diag[J₂(X1), J1 (A2), J1 (A3)].] 1 dim V₁, (A) = 3 and k = 2. [Answer: J = diag[J2(A1), J1 (A2), J₁ (A₂)], or J = diag[J₁ (A₁), J₁ (λ2), J2 (λ2)].]
1. Let A € M4(F) and let J € M₂(F) be a Jordan canonical form of A (you may assume A = J to do the questions below). Find J and the characteristic and minimal polynomials pÃ(™), m₁(x) of A for each case below, where A₁,..., Ak are the only distinct eigenvalues of A. Hint. See T8.7 and its General remark. k 1a. Σ₁ dim V₁, (A) = 4. [Answer: J is diagonal (5 cases).] i=1 1b. 1c. 1 dim Vx, (A) = 3 and k = 3. [Answer: diag[J₂(X1), J1 (A2), J1 (A3)].] 1 dim V₁, (A) = 3 and k = 2. [Answer: J = diag[J2(A1), J1 (A2), J₁ (A₂)], or J = diag[J₁ (A₁), J₁ (λ2), J2 (λ2)].]
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
![1. Let A € M4(F) and let J € M₂(F) be a Jordan canonical form of A
(you may assume A = J to do the questions below).
Find J and the characteristic and minimal polynomials PA(z), m₁(x) of A
for each case below, where A₁,..., Ak are the only distinct eigenvalues of A.
Hint. See T8.7 and its General remark.
1a.
₁ dim V₁₂ (A) = 4. [Answer: J is diagonal (5 cases).]
1b.
1 dim V₁₂ (A) = 3 and k = 3. [Answer: diag[J₂(X1), J1 (A2), J1 (A3)].]
1c. Σ₁ dim V₁, (A) = 3 and k = 2.
[Answer: J = diag[J₂(A1), J1 (A2), J1 (A2)], or J = diag[J₁ (A₁), J1 (A2), J2 (A2)].]
1d. Ei=1
dim Vx, (A) = 3 and k = 1. [Answer: J = diag[J₁ (X1), J1 (X1), J2 (1)].]
1e. Σ1 dim V₁₂ (A) = 2 & k = 2. [J = diag[J1 (A1), J3(A2)], or diag[J2 (A1), J2(A2)].]
1f.dim V₁, (A)=2&k=1. [J = diag[J₁ (A₁), J3(1)], or diag[J₂ (A1), J2 (A1)].]
k
i=1
1g. Σ dim V₁, (A) = 1
k
i=1
[Answer: J = J4(X₁).]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fc235f5-c6fd-4e1c-bac8-7d394d2e8123%2F79daa137-7a47-40a7-9182-76cb600be1ce%2Fr32g3oe_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let A € M4(F) and let J € M₂(F) be a Jordan canonical form of A
(you may assume A = J to do the questions below).
Find J and the characteristic and minimal polynomials PA(z), m₁(x) of A
for each case below, where A₁,..., Ak are the only distinct eigenvalues of A.
Hint. See T8.7 and its General remark.
1a.
₁ dim V₁₂ (A) = 4. [Answer: J is diagonal (5 cases).]
1b.
1 dim V₁₂ (A) = 3 and k = 3. [Answer: diag[J₂(X1), J1 (A2), J1 (A3)].]
1c. Σ₁ dim V₁, (A) = 3 and k = 2.
[Answer: J = diag[J₂(A1), J1 (A2), J1 (A2)], or J = diag[J₁ (A₁), J1 (A2), J2 (A2)].]
1d. Ei=1
dim Vx, (A) = 3 and k = 1. [Answer: J = diag[J₁ (X1), J1 (X1), J2 (1)].]
1e. Σ1 dim V₁₂ (A) = 2 & k = 2. [J = diag[J1 (A1), J3(A2)], or diag[J2 (A1), J2(A2)].]
1f.dim V₁, (A)=2&k=1. [J = diag[J₁ (A₁), J3(1)], or diag[J₂ (A1), J2 (A1)].]
k
i=1
1g. Σ dim V₁, (A) = 1
k
i=1
[Answer: J = J4(X₁).]
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