Diagonalize the matrix A = [1,1;-2,4] using a similarity transformation. Give the transformation matrices such that (C^−1)AC = D, where D is a diagonal matrix. (D, C) = ?
To diagonalize a matrix A, we need to find a diagonal matrix D and a similarity transformation matrix C such that (C^-1)AC = D.
Here's how we can find D and C:
1. Start by finding the eigenvalues of matrix A:
- The characteristic equation of A is det(A - λI) = 0, where I is the identity matrix and λ is the eigenvalue.
- Substitute the values of A into the equation and solve for λ. In this case, we have:
|1 - λ, 1| => (1 - λ)(4 - λ) - (-2)(1) = 0
|-2, 4 - λ|
- Simplifying, we get: (1 - λ)(4 - λ) + 2 = 0
=> λ^2 - 5λ + 6 = 0
- Solving this quadratic equation, we find the eigenvalues λ_1 = 3 and λ_2 = 2.
2. Next, find the corresponding eigenvectors for each eigenvalue:
- For λ_1 = 3:
- Solve the equation (A - 3I)v_1 = 0
=> |1 - 3, 1| |x| = |0|
|-2, 4 - 3| |y| |0|
=> |-2, 1| |x| = |0|
|-2, 1| |y| |0|
- Solving this system of equations, we get v_1 = [1, 2]
- For λ_2 = 2:
- Solve the equation (A - 2I)v_2 = 0
=> |-1, 1| |x| = |0|
|-2, 4 - 2| |y| |0|
=> |-1, 1| |x| = |0|
|-2, 2| |y| |0|
- Solving this system of equations, we get v_2 = [1, 1]
3. Now, construct matrix C using the eigenvectors as its columns:
- C = [v_1, v_2]
= [1, 1; 2, 1]
4. Calculate the inverse of C, C^-1:
- C^-1 = [1, -1; -2, 1]
5. Finally, compute D using the formula D = (C^-1)AC:
- Substitute the values of C, A, and C^-1 into the equation:
D = (C^-1)AC
= [1, -1; -2, 1] * [1, 1; -2, 4] * [1, 1; 2, 1]
= [3, 0; 0, 2]
So, the diagonal matrix D = [3, 0; 0, 2] and the similarity transformation matrix C = [1, 1; 2, 1].