Find the Eigenvalues

Problem

[[x,−1/3,−1/3],[−1/3,x,−1/3],[−1/3,−1/3,x]]

Solution

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

det(x−λ)=0

2. Simplify the determinant calculation by adding all rows to the first row to identify a common factor.

det(x−λ−2/3)=0

3. Factor out the common term (x−λ−2/3) from the first row.

(x−λ−2/3)*det(1)=0

4. Perform row operations to create zeros in the first row, specifically (C_2)−(C_1) and (C_3)−(C_1)

(x−λ−2/3)*det(1)=0

5. Calculate the determinant of the resulting upper triangular matrix by multiplying the diagonal entries.

(x−λ−2/3)*(x−λ+1/3)2=0

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

(λ_1)=x−2/3

(λ_2)=x+1/3

(λ_3)=x+1/3

Final Answer

λ=x−2/3,x+1/3,x+1/3

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