Find the Eigenvalues

Problem

[[1,0,1,0],[0,1,1,0],[0,0,1,1],[1,1,0,1]]

Solution

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

det(A−λ*I)=|[1−λ,0,1,0],[0,1−λ,1,0],[0,0,1−λ,1],[1,1,0,1−λ]|=0

2. Expand the determinant along the first row.

(1−λ)*|[1−λ,1,0],[0,1−λ,1],[1,0,1−λ]|+1*|[0,1−λ,0],[0,0,1],[1,1,1−λ]|=0

3. Calculate the 3×3 determinants.

(1−λ)*((1−λ)3+1)+1*(−(1−λ)*(−1))=0

4. Simplify the resulting polynomial expression.

(1−λ)4+(1−λ)+(1−λ)=0

(1−λ)4+2*(1−λ)=0

5. Factor the polynomial to find the roots.

(1−λ)*((1−λ)3+2)=0

6. Solve for λ by setting each factor to zero.

1−λ=0⇒(λ_1)=1

(1−λ)3=−2⇒1−λ=√(3,−2)

λ=1−√(3,−2)

7. Identify all three roots of the cubic part, including complex roots using the cube roots of −1

(λ_2)=1+√(3,2)

(λ_3)=1+√(3,2)*e(iπ/3)=1+√(3,2)*(1/2+i√(,3)/2)

(λ_4)=1+√(3,2)*e(−iπ/3)=1+√(3,2)*(1/2−i√(,3)/2)

Final Answer

λ={1,1+√(3,2),1−√(3,2)/2+i(√(3,2)√(,3))/2,1−√(3,2)/2−i(√(3,2)√(,3))/2}

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