Find the Eigenvectors/Eigenspace [[1.3,0.4],[-0.2,1.9]]

Problem

[[1.3,0.4],[−0.2,1.9]]

Solution

1. Identify the matrix A and the characteristic equation det(A−λ*I)=0 to find the eigenvalues.

A=[[1.3,0.4],[−0.2,1.9]]

2. Calculate the determinant of A−λ*I

det(1.3−λ)=(1.3−λ)*(1.9−λ)−(0.4)*(−0.2)

3. Expand the characteristic polynomial.

(1.3−λ)*(1.9−λ)+0.08=λ2−3.2*λ+2.47+0.08

λ2−3.2*λ+2.55=0

4. Solve for λ using the quadratic formula.

λ=(3.2±√(,(−3.2)2−4*(1)*(2.55)))/2

λ=(3.2±√(,10.24−10.2))/2

λ=(3.2±√(,0.04))/2

λ=(3.2±0.2)/2

(λ_1)=1.7,(λ_2)=1.5

5. Find the eigenvector for (λ_1)=1.7 by solving (A−1.7*I)*(v_1)=0

[[1.3−1.7,0.4],[−0.2,1.9−1.7]]*[[x],[y]]=[[0],[0]]

[[−0.4,0.4],[−0.2,0.2]]*[[x],[y]]=[[0],[0]]

−0.4*x+0.4*y=0⇒x=y

(v_1)=[[1],[1]]

6. Find the eigenvector for (λ_2)=1.5 by solving (A−1.5*I)*(v_2)=0

[[1.3−1.5,0.4],[−0.2,1.9−1.5]]*[[x],[y]]=[[0],[0]]

[[−0.2,0.4],[−0.2,0.4]]*[[x],[y]]=[[0],[0]]

−0.2*x+0.4*y=0⇒x=2*y

(v_2)=[[2],[1]]

Final Answer

Eigenspaces: *(E_1.7)=span*{[1],[1]},(E_1.5)=span*{[2],[1]}

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