Find the Eigenvectors/Eigenspace

Problem

[[1,2,6,8],[1,56,2,85],[16,7,9,3],[4,5,7,99]]

Solution

1. Set up the characteristic equation by subtracting λ from the diagonal elements of the matrix A and setting the determinant of A−λ*I to zero.

det(A−λ*I)=0

2. Calculate the determinant of the 4×4 matrix to find the characteristic polynomial.

det(1−λ)=0

3. Solve for the eigenvalues (λ_1),(λ_2),(λ_3),(λ_4) by finding the roots of the resulting quartic polynomial.

λ4−165*λ3+5606*λ2+10183*λ−517451=0

4. Approximate the eigenvalues using numerical methods, as this specific matrix does not yield simple integer roots.

(λ_1)≈121.37

(λ_2)≈55.45

(λ_3)≈−18.72

(λ_4)≈6.90

5. Find the eigenvectors for each λ by solving the homogeneous system (A−λ*I)*v=0 using Gaussian elimination.

(A−(λ_i)*I)*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

6. Determine the eigenspace for each eigenvalue, which is the span of the resulting linearly independent eigenvectors.

(E_(λ_i))=span{(v_i)}

Final Answer

Eigenvalues: *λ≈{121.37,55.45,−18.72,6.90}

Eigenspaces: *(E_(λ_i))=null*(A−(λ_i)*I)

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