Find the Eigenvalues

Problem

[[5,0,−1,0],[1,3,0,0],[2,−1,3,0],[4,−2,2,4]]

Solution

1. Set up the characteristic equation by finding the determinant of A−λ*I

det(A−λ*I)=|[5−λ,0,−1,0],[1,3−λ,0,0],[2,−1,3−λ,0],[4,−2,2,4−λ]|=0

2. Expand the determinant along the fourth column, which contains three zeros.

(4−λ)*|[5−λ,0,−1],[1,3−λ,0],[2,−1,3−λ]|=0

3. Calculate the 3×3 determinant using expansion along the first row.

(4−λ)*((5−λ)*((3−λ)2−0)−0+(−1)*(−1−2*(3−λ)))=0

4. Simplify the expression inside the brackets.

(4−λ)*((5−λ)*(3−λ)2−(−1−6+2*λ))=0

(4−λ)*((5−λ)*(9−6*λ+λ2)−(2*λ−7))=0

5. Expand the cubic polynomial further.

(4−λ)*(45−30*λ+5*λ2−9*λ+6*λ2−λ3−2*λ+7)=0

(4−λ)*(−λ3+11*λ2−41*λ+52)=0

6. Factor the cubic polynomial by testing possible roots. Testing λ=4

−(4)3+11*(4)2−41*(4)+52=−64+176−164+52=0

(4−λ)*(λ−4)*(−λ2+7*λ−13)=0

7. Solve for the remaining roots using the quadratic formula on −λ2+7*λ−13=0

λ=(−7±√(,7−4*(−1)*(−13)))/(2*(−1))

λ=(−7±√(,49−52))/(−2)

λ=(7±i√(,3))/2

8. List all eigenvalues found from the factors (4−λ) (λ−4) and the quadratic roots.

(λ_1)=4

(λ_2)=4

(λ_3)=(7+i√(,3))/2

(λ_4)=(7−i√(,3))/2

Final Answer

λ=4,4,(7±i√(,3))/2

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