Reduce the quadratic form Q(x)=x^2+y^2+z^2+2xy-4yz+6xz to diagonal form.
The symmetric matrix for Q is
(1 1 3)
(1 1 -2)
(3 -2 1)
Can you now reduce that to diagonal form?
To reduce the quadratic form Q(x) = x^2 + y^2 + z^2 + 2xy - 4yz + 6xz to diagonal form, we need to find the matrix representation of the quadratic form, then use a suitable matrix transformation to diagonalize it.
Step 1: Write the quadratic form in matrix representation.
Q(x) = x^2 + y^2 + z^2 + 2xy - 4yz + 6xz
The matrix representation of the quadratic form is:
Q(x) = [x, y, z] * [1 1 3;
1 1 -2;
3 -2 1] * [x;
y;
z]
Step 2: Diagonalize the matrix.
To diagonalize the matrix [1 1 3;
1 1 -2;
3 -2 1], we need to find its eigenvalues and corresponding eigenvectors.
a) Find the eigenvalues:
The eigenvalues λ satisfy the equation: |A - λI| = 0, where A is the matrix [1 1 3;
1 1 -2;
3 -2 1] and I is the identity matrix.
Using the equation (A - λI) |v⟩ = 0, where |v⟩ is the eigenvector corresponding to the eigenvalue λ, we will solve for the eigenvalues.
(A - λI) |v⟩ = 0 can be written as:
[1-λ 1 3 ] [x] [0]
x̂ = [1 1-λ -2 ] [y] = [0]
[3 -2 1-λ] [z] [0]
Expanding the determinant |A - λI| = 0 and solving for the eigenvalues λ, we have:
(1-λ)[(1-λ)(1-λ) - (-2)(3)] - [1(-2)(1-λ) + 1(3)(3) + 3(1)(1-λ)] = 0
Simplifying the above equation, we get:
(λ-2)(λ^2 - 5λ + 6) = 0
Solving λ^2 - 5λ + 6 = 0, we find two eigenvalues: λ1 = 2 and λ2 = 3.
b) Find the eigenvectors:
To find the eigenvectors, we substitute the eigenvalues back into the equation (A - λI) |v⟩ = 0 and solve for |v⟩.
For λ1 = 2:
(A - 2I) |v1⟩ = 0 can be written as:
[-1 1 3 ] [x] [0]
x̂ = [1 -1 -2 ] [y] = [0]
[3 -2 -1] [z] [0]
Simplifying the above equations, we have:
-x + y + 3z = 0
x - y - 2z = 0
3x - 2y - z = 0
Solving the above system of equations, we get the eigenvector corresponding to λ1 = 2 as:
|v1⟩ = [1;
1;
-1]
For λ2 = 3:
(A - 3I) |v2⟩ = 0 can be written as:
[-2 1 3 ] [x] [0]
x̂ = [1 -2 -2 ] [y] = [0]
[3 -2 -2] [z] [0]
Simplifying the above equations, we have:
-2x + y + 3z = 0
x - 2y - 2z = 0
3x - 2y - 2z = 0
Solving the above system of equations, we get the eigenvector corresponding to λ2 = 3 as:
|v2⟩ = [1;
1;
1]
Step 3: Diagonalize the matrix.
Now that we have found the eigenvalues and eigenvectors, we can diagonalize the matrix [1 1 3;
1 1 -2;
3 -2 1].
The diagonalized form of the matrix is:
D = [2 0 0;
0 3 0;
0 0 3]
Step 4: Transform back to quadratic form.
To transform the diagonalized matrix, D, back to quadratic form, we multiply it with the eigenvector matrix, P, and its inverse, P^(-1).
Let P = [1 1 1;
1 1 1;
-1 1 1], which is the matrix formed by the eigenvectors.
We have:
P * D * P^(-1) = [1 1 1;
1 1 1;
-1 1 1] * [2 0 0;
0 3 0;
0 0 3] * [1/3 -1/3 -1/3;
-1/3 2/3 -1/3;
-1/3 -1/3 2/3]
Simplifying the above expression, we get:
P * D * P^(-1) = [6 0 0;
0 2 0;
0 0 2]
Thus, the reduced diagonal form of the quadratic form Q(x) = x^2 + y^2 + z^2 + 2xy - 4yz + 6xz is Q(x) = 6x^2 + 2y^2 + 2z^2.
To reduce the quadratic form Q(x) = x^2 + y^2 + z^2 + 2xy - 4yz + 6xz to diagonal form, we need to find a scalar matrix A such that the transformed quadratic form Q'(x) = x'^T A x', where x' is the transformed vector.
The quadratic form Q(x) can be represented by a symmetric matrix called the coefficient matrix. The coefficient matrix contains the coefficients of the variables and their interactions. In this case, the coefficient matrix is:
A = [[1, 1, 3],
[1, 1, -2],
[3, -2, 1]]
To diagonalize the matrix A, we need to find a matrix P such that P^-1 * A * P is a diagonal matrix. Here's how we can proceed:
1. Find the eigenvalues of matrix A:
- Calculate the determinant of the matrix A - λ * I, where I is the identity matrix and λ is an eigenvalue.
- Set the determinant to zero and solve for λ to get the eigenvalues.
2. Find the eigenvectors corresponding to each eigenvalue:
- Substitute each eigenvalue into the equation A * v = λ * v, where v is the eigenvector.
- Solve the linear system of equations to find the eigenvector.
3. Assemble the matrix P:
- The matrix P is formed by taking the eigenvectors corresponding to the eigenvalues as its columns.
4. Diagonalize the matrix A:
- Calculate the inverse of P, denoted P^-1.
- Calculate the diagonal matrix D = P^-1 * A * P.
- The diagonal elements of D correspond to the eigenvalues, while the off-diagonal elements are all zero.
By following these steps, you will obtain the diagonal form of the quadratic form Q(x), where the off-diagonal elements will be zero.