ExampleLet1-44A =-6 5-54.The characteristic polynomial of A is CA(^) =-(A + 1)² (A – 1), so –1 is an eigenvalue with algebraic multiplicityand 1 is an eigenvalue withalgebraic multiplicityWithout further calculations, we may determine from the inequalities 1 < geo(A) < alg(A) that the geometric multiplicity of 1isFor the other eigenvalue, –1, all that the inequalities tell us is that 1 < geo(-1) <However, after some row operations, we find that thematrix A + I has rank 2, so the dimension of its null space isshowing that the geometric multiplicity of –1 isBy computing reduced row-echelon forms, we find that the eigenspace associated to 1 is spanned by1and the eigenspace associated to -1 is spanned by1|.This example illustrates the following important case of the theorem above.Important special case: If the algebraic multiplicity of an eigenvalue A is 1, so alg(A) = 1, then the inequality in the theorem above reduces to1< geo(A) < 1. This can only be true when geo(A) = 1.

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Answered: Example Let -- [1 -4 4 A = 2 -6 5. -5…

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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Problem 2: Suppose that the characteristic polynomial of some ma- trix A is found to be p(X) = ²(A – 1)*(A – 2)². Use this to answer the following a) What is the size of A? b) Is A invertible? c) How many eigenspaces does A have? Justify all your answers. NO JUSTIFICATION = NO CREDIT.
3 *. Sketch a graph of the eigenvectors and plot Solve the system of ODES given by i' = | , some sample trajectories. Is the origin an attractor, a repeller or a saddle point? -2.
Let A = Find the Characteristic Polynomial, and compute the Eigenvalues: They are (note i = v-1) is the iaginary "unit"): O p(A) = X² + 21 +1=A1 = -1, and A2 = -1, O p(A) = X² – 1=\1 = -1, and A2 = 1 O p(A) = X² – A +1=A1 = 1, and dz O p(A) = X² + 2A + 2 = A1 = -1+ i, and X2 = -1 - i O p(A) = X – 2A + 2 A1 = 1 + i, and A, = 1 – i O p(A) = X2 +1=\1 = i, and A2 = -i 1 W 000 000 F4 F3 E5
Let x(t) x₁ (t) x' (t) x₁ (t): = = If x (0) = x₂(t): = = = ] Put the eigenvalues in ascending order when you enter x₁ (t), x₂(t) below. exp( t) + exp( x₁(t) x₂(t) -27 x₁(t) + 12x₂(t) -56 x₁(t) + 25 x₂(t) 4 be a solution to the system of differential equations: -2 exp( find x(t). t) + exp( t) t)
1. Let A = has eigenvalues X₁ = 2 2 3 -1 -2 2 1 and X₂ 5. You can assume, and do NOT need to show, that A Compute algebraic and geometric multiplicities of X₁ and ₂.
1 Let P = -2 - Find p P. Deduce P, if it exists, from the result of -1 p' P. You can also calculate P 1 using the traditional approach but this will be slower. 2. Let A be a 3 x 3 matrix with the following eigen-pairs: 1 (1, ):(1, 0 ):G -2 ).
П Write the function f(x) = x (1-x) as an eigenfunction ex- pansion of the eigenfunctions corresponding to the Sturm-Liouville prob- lem y" + Xy = 0, 0 < x < y(0) = 0 y (H) = 0 πT
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
(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?
value problem. find the general solution of the homogeneous linear system and then solve the given initial
7. Let the 2 x 2 matrix A have real, distinct eigenvalues and μ. Suppose that an eigenvector of A is (1,0)" and an eigenvector of u is (-1,1)". Sketch the phase portraits of is x = Ax for the following cases: (1) 0 0. وچستان ستان ج

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