Find the Eigenvalues [[0,1],[-1, square root of 2]]

Problem

[[0,1],[−1,√(,2)]]

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=[[0,1],[−1,√(,2)]]

2. Set up the determinant for the characteristic polynomial by subtracting λ from the diagonal elements.

det(−λ)=0

3. Calculate the determinant by applying the formula a*d−b*c

(−λ)*(√(,2)−λ)−(1)*(−1)=0

4. Expand the expression to form a quadratic equation in terms of λ

−√(,2)*λ+λ2+1=0

λ2−√(,2)*λ+1=0

5. Apply the quadratic formula λ=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=−√(,2) and c=1

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

6. Simplify the terms under the square root.

λ=(√(,2)±√(,2−4))/2

λ=(√(,2)±√(,−2))/2

7. Express the result using the imaginary unit i where √(,−2)=i√(,2)

λ=(√(,2)±i√(,2))/2

8. Divide each term by 2 to find the final complex eigenvalues.

λ=√(,2)/2±(i√(,2))/2

Final Answer

λ=√(,2)/2+(i√(,2))/2,√(,2)/2−(i√(,2))/2

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