Find the Eigenvalues [[0.6,0.8],[0.8,-0.6]]

Problem

[[0.6,0.8],[0.8,−0.6]]

Solution

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

det(A−λ*I)=0

2. Write the determinant expression for the given matrix.

|[0.6−λ,0.8],[0.8,−0.6−λ]|=0

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

(0.6−λ)*(−0.6−λ)−(0.8)*(0.8)=0

4. Simplify the quadratic expression.

−(0.6−λ)*(0.6+λ)−0.64=0

−(0.36−λ2)−0.64=0

−0.36+λ2−0.64=0

λ2−1=0

5. Solve for λ by taking the square root of both sides.

λ2=1

λ=±1

Final Answer

λ=1,−1

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