Find the Eigenvalues A=[[cos(o),-sin(o)],[sin(o),cos(o)]]

Problem

A=[[cos(θ),−sin(θ)],[sin(θ),cos(θ)]]

Solution

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

det(A−λ*I)=0

2. Subtract λ from the diagonal elements of the matrix A

A−λ*I=[[cos(θ)−λ,−sin(θ)],[sin(θ),cos(θ)−λ]]

3. Calculate the determinant of the resulting 2×2 matrix.

det(A−λ*I)=(cos(θ)−λ)*(cos(θ)−λ)−(−sin(θ))*(sin(θ))

4. Expand the expression and simplify using the Pythagorean identity sin2(θ)+cos2(θ)=1

λ2−2*λ*cos(θ)+cos2(θ)+sin2(θ)=0

λ2−2*λ*cos(θ)+1=0

5. Apply the quadratic formula to solve for λ where a=1 b=−2*cos(θ) and c=1

λ=(−(−2*cos(θ))±√(,(−2*cos(θ))2−4*(1)*(1)))/(2*(1))

6. Simplify the radical using the identity 1−cos2(θ)=sin2(θ) and the imaginary unit i

λ=(2*cos(θ)±√(,4*cos2(θ)−4))/2

λ=(2*cos(θ)±√(,−4*(1−cos2(θ))))/2

λ=(2*cos(θ)±2*i*sin(θ))/2

7. Divide by 2 to find the final eigenvalues.

λ=cos(θ)±i*sin(θ)

Final Answer

λ=cos(θ)±i*sin(θ)

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