Find the Eigenvectors/Eigenspace [[1,-2],[-3,6]]

Problem

[[1,−2],[−3,6]]

Solution

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

A=[[1,−2],[−3,6]]

2. Calculate the determinant of A−λ*I

det(1−λ)=(1−λ)*(6−λ)−(−2)*(−3)

3. Expand and simplify the characteristic polynomial.

(1−λ)*(6−λ)−6=λ2−7*λ+6−6

λ2−7*λ=0

4. Solve for the eigenvalues by factoring.

λ*(λ−7)=0

(λ_1)=0,(λ_2)=7

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

[[1,−2],[−3,6]]*[[(x_1)],[(x_2)]]=[[0],[0]]

(x_1)−2*(x_2)=0⇒(x_1)=2*(x_2)

(v_1)=[[2],[1]]

6. Find the eigenvector for (λ_2)=7 by solving (A−7*I)*v=0

[[1−7,−2],[−3,6−7]]=[[−6,−2],[−3,−1]]

−3*(x_1)−(x_2)=0⇒(x_2)=−3*(x_1)

(v_2)=[[1],[−3]]

Final Answer

Eigenspace for *λ=0:span*{[2],[1]},Eigenspace for *λ=7:span*{[1],[−3]}

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