To determine the nature, index, and signature of the given quadratic form $Q(x_1, x_2, x_3) = x_1^2 + 5x_2^2 + x_3^2 + 2x_2x_3 + 6x_3x_1 + 2x_1x_2$, we can follow these steps:

1. Matrix Representation of the Quadratic Form

A quadratic form can be written as:

$Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}$

where $\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$ and $A$ is the symmetric matrix representing the quadratic form. The general structure of $A$ is determined by the coefficients of the terms $x_i^2$ and the mixed terms $x_ix_j$.

The symmetric matrix $A$ for this quadratic form is:

1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{pmatrix}$$ Here, the diagonal elements correspond to the coefficients of $$x_1^2, x_2^2, x_3^2$$, and the off-diagonal elements correspond to the coefficients of the mixed terms (divided by 2, because each mixed term is counted twice in the quadratic form). ### 2. Eigenvalues of the Matrix To determine the nature, index, and signature, we need the **eigenvalues** of matrix $$A$$. The signature depends on the number of positive, negative, and zero eigenvalues. We solve for the eigenvalues of $$A$$ by solving the characteristic equation: $$\det(A - \lambda I) = 0$$ The determinant of $$A - \lambda I$$ is: $$\det\left(\begin{pmatrix} 1-\lambda & 1 & 3 \\ 1 & 5-\lambda & 1 \\ 3 & 1 & 1-\lambda \end{pmatrix}\right) = 0$$ Solving this cubic equation gives the eigenvalues. After computing, the eigenvalues are approximately: $$\lambda_1 = 6, \quad \lambda_2 = 3, \quad \lambda_3 = -2$$ ### 3. Nature, Index, and Signature - **Nature**: The quadratic form is **indefinite**, as it has both positive and negative eigenvalues. - **Signature**: The signature of the quadratic form is $$(2, 1)$$, which means it has 2 positive eigenvalues and 1 negative eigenvalue. - **Index**: The **index** of the quadratic form is the number of negative eigenvalues, so the index is $$1$$. ### Conclusion: - **Nature**: Indefinite - **Index**: 1 - **Signature**: $$(2, 1)$$ Would you like more details on how the eigenvalues were calculated or further clarification? Here are some follow-up questions for deeper understanding: 1. How does the signature of a quadratic form relate to its definiteness? 2. What is the significance of the index of a quadratic form in applications? 3. How can the matrix representation of a quadratic form help in finding its nature? 4. What are the implications of having an indefinite quadratic form in physics or geometry? 5. How can diagonalization simplify understanding the quadratic form? **Tip**: Diagonalizing the matrix of a quadratic form can help quickly identify its nature by converting it into a simpler form, where the eigenvalues clearly indicate positivity or negativity.