Consider the following.3 02(a) Compute the characteristic polynomial of A.det (4-21)-(2-2)(2+1)(2-2)2₁A =2₂ =0(b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.)(E)-2₂xhas eigenspace spanhas eigenspace spanhas eigenspace span(smallest λ-value)--(largest λ-value)

Eingvalues found by finding characteristics equation.

Answered: Consider the following. (a) Compute the…

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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The characteristic polynomial of A is p(λ) = λ3−3λ2+4. Find its eigenvalues.
For (a) and (b), 2 -1 [A] = -1 2 (a) What are the eigenvalues of A? (b) For each eigenvalue, find a nonzero eigenvector. [1/3] [1/4] (c) Suppose that the matrix B has the eigenvalues 1 and -6, with eigenvectors and respectively. Calculate Q(t).N(0)'. (Q: fundamental matrix for u’=B.u) (d) What is the solution to u'=B.u with u(0)=| ?
Verify that λ; is an eigenvalue of A and that x; is a corresponding eigenvector. 2₁9, x₁ (1, 0) = - [8. AX1 = AX2 A = = 9 0 0-9 = 22-9, x₂ = (0, 1) = = ↓ 1 = 9 0 -9 22x2 ~-(:D)- E= -0)-*-*2 1 0-9 ↓1 •[:] · = 2₁x1
Consider the following. A₁ = A = (a) Compute the characteristic polynomial of A. det (A-AI)=-23-32² d₂= 5 50 0 -9 1 1 1 (b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.) ^3 = -1 0 (LF has eigenspace span has eigenspace span has eigenspace span ↓1 1 (c) Compute the algebraic and geometric multiplicity of each eigenvalue. A₁ has algebraic multiplicity and geometric multiplicity λ₂ has algebraic multiplicity and geometric multiplicity 13 has algebraic multiplicity and geometric multiplicity (smallest λ-value) (largest λ-value)
ul T-Mobile 2:44 AM 37% Let A = Find the Characteristic Polynomial, and compute the Eigenvalues: They are (note i = v-1) is the imaginary "unit"): O p(A) = X2+ 21 +1=Aj = -1, and A2 = -1, O p(A) = X2-1 =-1, and A2 = 1 O p(A) = X² – 24 +1=A, = 1, and A2 = 1 O p(A) = X² +2A + 2 A1 = -1+i, and A2 = -1-i O p(A) = X2 - 2A +2 A1 = 1+i, and A2 = 1-i O p(A) = X2 + 1=A1 = i, and X2 = -i F3 O00 F4 F5 F7
Determine the particular solution z(x, y) of the partial differential equation a²z 10x³y + 12x?y2 ax2 subject to the boundary conditions z = y and az = y? ax when x = 1.
(8) Show that the constant term of the characteristic polynomial |21 – A| is ±JA|. Using this fact, consider the case here A = 0 is an eigenvalue of A. What can you conclude about the invertibility of A, and why?
2: Let 10 -1 -1 A = 0. 10 10 -1 -1 10 (a) Find the characteristic polynomial of A. (b) Find the dimension of the eigenspace corresponding to the largest eigenvalue.
A = [ 3² -7] ₁ 1. Let A = (a) By finding the eigenvalues and corresponding eigenvectors, give a diagonalization of A. (b) Without computing directly, find A¹ using the diagonalization from part (a).
Consider the following. (a) Compute the characteristic polynomial of A. det(A - AI) = -2³ +72² +172 +9 A₁ = A = (b) Compute the eigenvalues and bases of the corresponding eigenspaces of A. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.) ^₂ = 4 05 6-1 6 5 04 23 = has eigenspace span has eigenspace span has eigenspace span ↓ 1 (c) Compute the algebraic and geometric multiplicity of each eigenvalue. A₁ has algebraic multiplicity and geometric multiplicity ₂ has algebraic multiplicity and geometric multiplicity 13 has algebraic multiplicity and geometric multiplicity (smallest λ-value) (largest λ-value)
Let A (a) Determine the eigenvalues X₁ and ₂ of A where X₁
Verify that 1; is an eigenvalue of A and that X¡ is a corresponding eigenvector. 11 = 5, x1 = (1, 0, 0) 12 = 3, x2 = (1, 2, 0) 23 = 4, x3 = (-4, 1, 1) 5 -1 5 A = 3 1 0 4 5 -1 5 1 1 Ax1 5 0 = 11X1 3 1 0 4 5 -1 5 1 1 Ax2 3 1 2 3 2 0 4 5 -1 5 4 Ax3 3 1 1 1 23X3 0 4

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