Find the Eigenvalues [[0.5,0.5],[0.5,0.5]]

Problem

[[0.5,0.5],[0.5,0.5]]

Solution

1. Identify the matrix A and the characteristic equation det(A−λ*I)=0 where I is the identity matrix and λ represents the eigenvalues.

A=[[0.5,0.5],[0.5,0.5]]

2. Set up the characteristic equation by subtracting λ from the diagonal elements and calculating the determinant.

det(0.5−λ)=0

3. Expand the determinant using the formula (a−λ)*(d−λ)−b*c

(0.5−λ)*(0.5−λ)−(0.5)*(0.5)=0

4. Simplify the quadratic equation.

(0.5−λ)2−0.25=0

0.25−λ+λ2−0.25=0

λ2−λ=0

5. Factor the quadratic equation to find the roots.

λ*(λ−1)=0

6. Solve for λ to determine the eigenvalues.

(λ_1)=0

(λ_2)=1

Final Answer

λ=0,1

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