1. Let A € M4(F) and let J € M₁(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(™), MÂ(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. dim V, (A) = 4. [Answer: J is diagonal (5 cases).] i=1 1b. 1 dim V₁, (A) = 3 and k = 3. i=1 1c. 1 dim V₁₂ (A) = 3 and k = 2. [Answer: J = diag[J₂ (A₁), J₁ (A2), J₁ (A2)], or J = diag[J₁ (A₁), J₁ (A2), J2(A2)].] 1d. - dim Vx, (A) = 3 and k = 1. [Answer: J = diag[J₁ (A₁), J₁ (A₁), J2(A1)].] =1 1e. i=1 1f. dim V(A) = 2 & k = 2. [J = diag[J1(A1), J3(A2)], or diag[J2 (A1), J2(A2)].] 1 dim V₁, (A) = 2 & k = 1. [J = diag[J₁ (A₁), J3(A1)], or diag[J2(A1), J2(A1)].] 1 dim V₁, (A) = 1 [Answer: J = J4(X1).] 1g. [Answer: diag[J2 (A1), J1 (A2), J1 (A3)].]
1. Let A € M4(F) and let J € M₁(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(™), MÂ(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. dim V, (A) = 4. [Answer: J is diagonal (5 cases).] i=1 1b. 1 dim V₁, (A) = 3 and k = 3. i=1 1c. 1 dim V₁₂ (A) = 3 and k = 2. [Answer: J = diag[J₂ (A₁), J₁ (A2), J₁ (A2)], or J = diag[J₁ (A₁), J₁ (A2), J2(A2)].] 1d. - dim Vx, (A) = 3 and k = 1. [Answer: J = diag[J₁ (A₁), J₁ (A₁), J2(A1)].] =1 1e. i=1 1f. dim V(A) = 2 & k = 2. [J = diag[J1(A1), J3(A2)], or diag[J2 (A1), J2(A2)].] 1 dim V₁, (A) = 2 & k = 1. [J = diag[J₁ (A₁), J3(A1)], or diag[J2(A1), J2(A1)].] 1 dim V₁, (A) = 1 [Answer: J = J4(X1).] 1g. [Answer: diag[J2 (A1), J1 (A2), J1 (A3)].]
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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Please help with the last subpart.
![1. Let A € M4(F) and let J € Mn(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(x), mд(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. E dim V¡ (A) = 4. [Answer: J is diagonal (5 cases).]
i=1
i=1
1b. Σ₁ dim V₁, (A) = 3 and k = 3.
1c. 1 dim V₂ (A) = 3 and k = 2.
[Answer: J = diag[J2(A1), J1 (A2), J₁ (A₂)], or J = diag[J1 (A1), J1 (A2), J2 (2)].]
1d. ₁ dim V₁₂ (A) = 3 and k = 1. [Answer: J = diag[J₁ (A1), J₁ (A1), J2(X1)].]
1e. E dim V₁₂ (A) = 2 & k = 2. [J = diag[J1 (X1), J3(A2)], or diag[J2(X1), J2(X2)].]
1f. Σ1 dim V₁₂ (A) = 2 & k = 1. [J = diag[J1 (A1), J3(A1)], or diag[J₂ (A1), J2 (1)].]
1g. - dim Vx, (A) = 1 [Answer: J = J4(X1).]
i=1
=1
[Answer: diag[J2(X1), J1 (A2), J1 (X3)].]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fc235f5-c6fd-4e1c-bac8-7d394d2e8123%2Ff886b9d0-eaea-4cdd-95c5-cf7121eaf8c3%2Fum9k3x7_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let A € M4(F) and let J € Mn(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(x), mд(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. E dim V¡ (A) = 4. [Answer: J is diagonal (5 cases).]
i=1
i=1
1b. Σ₁ dim V₁, (A) = 3 and k = 3.
1c. 1 dim V₂ (A) = 3 and k = 2.
[Answer: J = diag[J2(A1), J1 (A2), J₁ (A₂)], or J = diag[J1 (A1), J1 (A2), J2 (2)].]
1d. ₁ dim V₁₂ (A) = 3 and k = 1. [Answer: J = diag[J₁ (A1), J₁ (A1), J2(X1)].]
1e. E dim V₁₂ (A) = 2 & k = 2. [J = diag[J1 (X1), J3(A2)], or diag[J2(X1), J2(X2)].]
1f. Σ1 dim V₁₂ (A) = 2 & k = 1. [J = diag[J1 (A1), J3(A1)], or diag[J₂ (A1), J2 (1)].]
1g. - dim Vx, (A) = 1 [Answer: J = J4(X1).]
i=1
=1
[Answer: diag[J2(X1), J1 (A2), J1 (X3)].]
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