Find the Eigenvalues [[1,1],[-1,1]]

Problem

[[1,1],[−1,1]]

Solution

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

det(A−λ*I)=0

2. Write the determinant for the 2×2 matrix.

|[1−λ,1],[−1,1−λ]|=0

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

(1−λ)*(1−λ)−(1)*(−1)=0

4. Simplify the quadratic equation by distributing the terms and combining like terms.

(1−2*λ+λ2)+1=0

λ2−2*λ+2=0

5. Apply the quadratic formula λ=(−b±√(,b2−4*a*c))/(2*a) to solve for λ

λ=(−(−2)±√(,(−2)2−4*(1)*(2)))/(2*(1))

6. Simplify the expression under the square root and solve for the complex roots.

λ=(2±√(,4−8))/2

λ=(2±√(,−4))/2

λ=(2±2*i)/2

λ=1±i

Final Answer

λ=1+i,1−i

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