2. Determine the Jordan Canonical Form of A if the characteristic polynomialof A is given by (1-A)4(3-X)² with dim(Ker (A-1)) = 2 and dim(Ker (A-31)) = 1.

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Answered: 2. Determine the Jordan Canonical Form…

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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a1 a2 diag(a1, a2, ... , An) An Question 1 b3x3 + b2x2 + b1x + bo. Suppose that A = diag(a1, a2, a3) and suppose that the characteristic polynomial CA(x) = (1) Show that b3 = 1, b2 = -Tr(A) i.e. the trace of A, and bo = -det(A). (2) Give an example of C = diag(cı, c2, c3) and D = diag(dı, d2, d3) where Tr(C) = Tr(D), det(C) = det (D), but C is not similar to D.
If A = (aij) = M3 (K) for a field K then the characteristic polynomial of A has the form p₁(x) = - det(A) + C(A)x − Tr(A)x² + x³ where the coefficient of a can be computed as C(A) = a11@22 + a22a33 + a11933 a12a21a13a31 - a23a32 (a) is it necessarily true that if A is invertible then C(A¯¹) = C(A)? (b) is it necessarily true that if det (A) = Tr(A) = 0 then A³ = -C(A)A?
10. Consider the irreducible polynomial f = X' + X³ + X² + X + 1 in Z[X]. Let a = [X] in the ring Z[X]/(f). Express (a + 1)(a² + 1) in the form aza + aza² + aa + ao where a; E Z.
Let A E M₁(R) be such that A is not invertible. Which of the following cannot be the characteristic polynomial of A? Explain your answer. p(t) = 1¹-1², q(t) = t¹-1, r(t) = t4
Section 29 number 17

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2. Determine the Jordan Canonical Form of A if the characteristic polynomial of A is given by (1-A)4(3-A)2 with dim(Ker(A-1)) = 2 and dim(Ker(A- 31)) = 1.