1d.dim V₁, (A) = 3 and k = 1. [Answer: J = = diag[J₁ (A₁), J₁ (A1), J2(X1)].] 1e. E dim V₁, (A) = 2 & k = 2. [J = diag[J1(A1), J3(A2)], or diag[J2 (A1), J2(A2)].] i=1 1f. Σ1 dim Vx, (A) = 2 & k = 1. [J = diag[J1 (A1), J3(A1)], or diag[J2(A1), J2(A1)].] vi=1
1d.dim V₁, (A) = 3 and k = 1. [Answer: J = = diag[J₁ (A₁), J₁ (A1), J2(X1)].] 1e. E dim V₁, (A) = 2 & k = 2. [J = diag[J1(A1), J3(A2)], or diag[J2 (A1), J2(A2)].] i=1 1f. Σ1 dim Vx, (A) = 2 & k = 1. [J = diag[J1 (A1), J3(A1)], or diag[J2(A1), J2(A1)].] vi=1
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 help with subparts d, e, f.
![1. Let A € M₁(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 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. 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
1d.
1e. Σ₁1 dim V₂ (A) = 2 & k = 2. [J = diag[J1 (X1), J3(A2)], or diag[J2 (A1), J2(X2)].]
i=1
=1
1f. Σ1 dim V₁; (A) = 2 & k = 1. [J = diag[J1 (A1), J3(A1)], or diag[J₂ (A1), J2(A1)].]
1g. Σdim V₁₂ (A) = 1
[Answer: J = J4(X1).]
[Answer: diag[J2(X1), J1 (A2), J1 (X3)].]
(A₂)], or J = diag[J1 (A1), J1 (A2), J2 (2)].]
₁dim V₁, (A) = 3 and k = 1. [Answer: J = diag[J₁ (A1), J₁ (A1), J2(X1)].]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fc235f5-c6fd-4e1c-bac8-7d394d2e8123%2Fae0e4358-1b72-4b3e-b8f4-9942e24f76fe%2Fc0mr5z_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let A € M₁(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 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. 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
1d.
1e. Σ₁1 dim V₂ (A) = 2 & k = 2. [J = diag[J1 (X1), J3(A2)], or diag[J2 (A1), J2(X2)].]
i=1
=1
1f. Σ1 dim V₁; (A) = 2 & k = 1. [J = diag[J1 (A1), J3(A1)], or diag[J₂ (A1), J2(A1)].]
1g. Σdim V₁₂ (A) = 1
[Answer: J = J4(X1).]
[Answer: diag[J2(X1), J1 (A2), J1 (X3)].]
(A₂)], or J = diag[J1 (A1), J1 (A2), J2 (2)].]
₁dim V₁, (A) = 3 and k = 1. [Answer: J = diag[J₁ (A1), J₁ (A1), J2(X1)].]
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