https://file.aiquickdraw.com/mathbot/file/17267767595603zgszis9.jpg

The quadratic form presented in the image is:

\[
2x^2 + 2y^2 + 2z^2 - 2xy + 2xz - 2yz
\]

To reduce this quadratic form into its canonical form, we will follow these steps:

### 1. Represent the Quadratic Form Mathematically
The given quadratic form can be written in matrix form as:

\[
Q(x, y, z) = [x \ y \ z] \begin{bmatrix} a & b & c \\ b & d & e \\ c & e & f \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}
\]

From the quadratic form \( 2x^2 + 2y^2 + 2z^2 - 2xy + 2xz - 2yz \), we can deduce the coefficients:

- Coefficient of \(x^2\) is 2 → \( a = 2 \)
- Coefficient of \(y^2\) is 2 → \( d = 2 \)
- Coefficient of \(z^2\) is 2 → \( f = 2 \)
- Coefficient of \(xy\) is \(-2\) → \( b = -1 \)
- Coefficient of \(xz\) is \(2\) → \( c = 1 \)
- Coefficient of \(yz\) is \(-2\) → \( e = -1 \)

So, the symmetric matrix \( A \) corresponding to this quadratic form is:

\[
A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}
\]

### 2. Diagonalize the Matrix
Next, we will diagonalize the matrix \( A \) to express the quadratic form in canonical form. This involves finding the eigenvalues and eigenvectors of the matrix.

I will calculate the eigenvalues and eigenvectors to proceed with the diagonalization.

Let me compute these values.The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = 1, \quad \lambda_2 = 4, \quad \lambda_3 = 1
\]

The corresponding eigenvectors (normalized) are:

\[
v_1 = \begin{bmatrix} -0.8165 \\ -0.4082 \\ 0.4082 \end{bmatrix}, \quad v_2 = \begin{bmatrix} -0.5774 \\ 0.5774 \\ -0.5774 \end{bmatrix}, \quad v_3 = \begin{bmatrix} 0.3810 \\ 0.8159 \\ 0.4349 \end{bmatrix}
\]

### 3. Canonical Form
The quadratic form can be rewritten as:

\[
Q(x, y, z) = \lambda_1 x'^2 + \lambda_2 y'^2 + \lambda_3 z'^2
\]

Substituting the eigenvalues:

\[
Q(x, y, z) = x'^2 + 4y'^2 + z'^2
\]

This is the canonical form of the given quadratic form.

### 4. Nature, Index, Rank, and Signature
- **Nature**: Since all eigenvalues are non-negative, the quadratic form is positive semi-definite.
- **Rank**: The rank is the number of non-zero eigenvalues. Here, all eigenvalues are non-zero, so the rank is 3.
- **Index**: The number of positive eigenvalues. Here, all eigenvalues are positive, so the index is 3.
- **Signature**: The signature is the number of positive, zero, and negative eigenvalues. Here, it is (3, 0, 0), meaning three positive, zero negative, and zero zero eigenvalues.

Would you like more details or explanations?

Here are some follow-up questions to expand on this topic:

1. What are the properties of positive semi-definite matrices in general?
2. How does diagonalization help in simplifying quadratic forms?
3. Can you always diagonalize a symmetric matrix?
4. What is the physical or geometrical interpretation of the signature of a quadratic form?
5. How are eigenvalues related to the stability of systems?

**Tip:** When dealing with quadratic forms, diagonalization is key to simplifying them, especially when identifying their nature, signature, and index.

Reduce the quadratic form 2x² + 2y² + 2z² - 2xy + 2xz - 2yz into a canonical form and hence determine its nature, index, rank, and signature.

Learn how to reduce the quadratic form 2x² + 2y² + 2z² - 2xy + 2xz - 2yz into its canonical form using diagonalization. This involves calculating eigenvalues and eigenvectors to determine the nature, index, rank, and signature. Suitable for undergraduate students in linear algebra.

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