Find the Eigenvalues

Problem

[[0.87,0.02,0.02],[0.05,0.9,0.03],[0.08,0.08,0.95]]

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(A−λ*I)=0

2. Write the determinant expression for the given matrix.

|[0.87−λ,0.02,0.02],[0.05,0.9−λ,0.03],[0.08,0.08,0.95−λ]|=0

3. Expand the determinant along the first row.

(0.87−λ)*|[0.9−λ,0.03],[0.08,0.95−λ]|−0.02*|[0.05,0.03],[0.08,0.95−λ]|+0.02*|[0.05,0.9−λ],[0.08,0.08]|=0

4. Calculate the 2×2 determinants.

(0.87−λ)*((0.9−λ)*(0.95−λ)−0.0024)−0.02*(0.05*(0.95−λ)−0.0024)+0.02*(0.004−0.08*(0.9−λ))=0

5. Simplify the polynomial expression.

(0.87−λ)*(λ2−1.85*λ+0.8526)−0.02*(0.0451−0.05*λ)+0.02*(0.08*λ−0.068)=0

6. Combine like terms to form the cubic characteristic polynomial.

−λ3+2.72*λ2−2.4594*λ+0.7394=0

7. Solve for the roots of the polynomial. Note that the sum of the columns is 1 for every column (0.87+0.05+0.08=1 etc.), which implies (λ_1)=1 is an eigenvalue.

−(λ−1)*(λ2−1.72*λ+0.7394)=0

8. Apply the quadratic formula to find the remaining eigenvalues from λ2−1.72*λ+0.7394=0

λ=(1.72±√(,1.72−4*(0.7394)))/2

9. Evaluate the square root and solve for (λ_2) and (λ_3)

λ=(1.72±√(,2.9584−2.9576))/2

λ=(1.72±√(,0.0008))/2

λ≈(1.72±0.02828)/2

10. Finalize the values.

(λ_2)≈0.8741

(λ_3)≈0.8459

Final Answer

λ=1,0.8741,0.8459

Want more problems? Check here!