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) = ?
The eigenvectors of A are (1,2) and (1,1)
So, C = [1,1;2,1] and C^-1=[-1,1;2,-1]
C^-1 A C = [3,0;0,2]
See
http://www.wolframalpha.com/input/?i=%28inverse+{{1,1},{2,1}}%29*{{1,1},{-2,4}}*{{1,1},{2,1}}
To diagonalize a matrix A using a similarity transformation, we need to find a matrix C such that (C^−1)AC = D, where D is a diagonal matrix. The transformation matrices (D, C) can be found by following these steps:
Step 1: Find the eigenvalues of matrix A.
- Find the characteristic polynomial by subtracting the eigenvalue λ from the diagonal elements of A and taking the determinant.
- Set the characteristic polynomial equal to zero and solve for λ to find the eigenvalues.
For the matrix A = [1,1; -2,4], we can find the eigenvalues as follows:
det(A - λI) = 0
|1-λ, 1| = λ(4-λ) - (-2)(1)
|-2, 4 - λ|
(1-λ)(4-λ) - (-2)(-2) = (λ - 3)(λ - 2) = 0
Solving λ - 3 = 0 and λ - 2 = 0, we get eigenvalues λ1 = 3 and λ2 = 2.
Step 2: Find the eigenvectors corresponding to the eigenvalues.
- For each eigenvalue, substitute it back into A - λI and solve the resulting homogeneous system of linear equations to find the corresponding eigenvector(s).
For eigenvalue λ1 = 3,
(A - 3I)x = 0
|1-3, 1| |x1| = |0|
|-2, 4-3| |x2| |0|
The resulting system of equations is:
-2x1 + x2 = 0
x1 + x2 = 0
Solving these equations, we find a non-zero eigenvector corresponding to λ1 = 3 as x1 = -1 and x2 = 2.
For eigenvalue λ2 = 2,
(A - 2I)x = 0
|1-2, 1| |x1| = |0|
|-2, 4-2| |x2| |0|
The resulting system of equations is:
-x1 + x2 = 0
-2x1 + 2x2 = 0
Solving these equations, we find a non-zero eigenvector corresponding to λ2 = 2 as x1 = 1 and x2 = 1.
Step 3: Form the matrix C using the eigenvectors as columns.
- Matrix C is formed by stacking the eigenvectors as columns.
Let's form matrix C:
C = [x1_λ1, x1_λ2; x2_λ1, x2_λ2]
= [-1, 1; 2, 1]
Step 4: Form the diagonal matrix D.
- Matrix D is formed by placing the eigenvalues along the diagonal.
Let's form matrix D:
D = [λ1, 0; 0, λ2]
= [3, 0; 0, 2]
Therefore, the transformation matrices are (D, C) = ([3, 0; 0, 2], [-1, 1; 2, 1]).