Question

Express each of the following quadratic forms as a matrix product involving a symmetric coefficient matrix: (a) \(q=3 u^{2}-4 u v+7 v^{2}\) (b) \(q=u^{2}+7 u v+3 v^{2}\) (c) \(q=8 u v-u^{2}-31 v^{2}\) (d) \(q=6 x y-5 y^{2}-2 x^{2}\) (e) \(q=3 u_{1}^{2}-2 u_{1} u_{2}+4 u_{1} u_{3}+5 u_{2}^{2}+4 u_{3}^{2}-2 u_{2} u_{3}\) (f) \(q=-u^{2}+4 u v-6 u w-4 v^{2}-7 w^{2}\)

Short answer

Quadratic forms are expressed as matrix products: (a) \(\begin{pmatrix} 3 & -2 \\ -2 & 7 \end{pmatrix} \) (b) \(\begin{pmatrix} 1 & 3.5 \\ 3.5 & 3 \end{pmatrix}\) (c) \(\begin{pmatrix} -1 & 4 \\ 4 & -31 \end{pmatrix} \) (d) \(\begin{pmatrix} -2 & 3 \\ 3 & -5 \end{pmatrix}\) (e) \(\begin{pmatrix} 3 & -1 & 2 \\ -1 & 5 & -1 \\ 2 & -1 & 4 \end{pmatrix} \) (f) \(\begin{pmatrix} -1 & 2 & -3 \\ 2 & -4 & 0 \\ -3 & 0 & -7 \end{pmatrix} \).

Step-by-step solution

Understand the Quadratic Form

A quadratic form in variables (e.g., \(u, v\)) can be expressed as a matrix product \( \mathbf{x}^T \mathbf{A} \mathbf{x} \), where \( \mathbf{x} \) is a column vector of the variables, and \( \mathbf{A} \) is a symmetric coefficient matrix.

Write Each Quadratic as a Expression

For each part, collect terms involving the same variables:(a) \( q = 3u^2 - 4uv + 7v^2 \)(b) \( q = u^2 + 7uv + 3v^2 \)(c) \( q = 8uv - u^2 - 31v^2 \)(d) \( q = 6xy - 5y^2 - 2x^2 \)(e) \( q = 3u_1^2 - 2u_1u_2 + 4u_1u_3 + 5u_2^2 + 4u_3^2 - 2u_2u_3 \)(f) \( q = -u^2 + 4uv - 6uw - 4v^2 - 7w^2 \)

Identify Coefficient Matrix Components

For each term in the quadratic expressions, assign coefficients to the symmetric matrix. The diagonal entries of the matrix are the coefficients of \( u^2, v^2, \) etc. Off-diagonal terms are half the coefficients of the cross product terms \( uv, xy, etc. \) since they will contribute equally from two positions in the matrix product.

Construct Symmetric Matrices for Each Quadratic

Using the rules from Step 3, create the matrices:(a) \( \begin{pmatrix} 3 & -2 \ -2 & 7 \end{pmatrix} \)(b) \( \begin{pmatrix} 1 & 3.5 \ 3.5 & 3 \end{pmatrix} \)(c) \( \begin{pmatrix} -1 & 4 \ 4 & -31 \end{pmatrix} \)(d) \( \begin{pmatrix} -2 & 3 \ 3 & -5 \end{pmatrix} \)(e) \( \begin{pmatrix} 3 & -1 & 2 \ -1 & 5 & -1 \ 2 & -1 & 4 \end{pmatrix} \)(f) \( \begin{pmatrix} -1 & 2 & -3 \ 2 & -4 & 0 \ -3 & 0 & -7 \end{pmatrix} \)

Verify the Symmetric Matrix Forms

Verify each symmetric matrix by expanding \( \mathbf{x}^T \mathbf{A} \mathbf{x} \) for each matrix and checking if it yields the original quadratic expressions. If discrepancies arise, re-check your calculations until the expression and matrix product align.

Key concepts

Symmetric Matrix

A symmetric matrix is a square matrix that remains unchanged when its rows and columns are swapped. This concept is essential in representing quadratic forms efficiently. In simple terms, a matrix \(A\) is symmetric if \(A = A^T\), where \(A^T\) means the transpose of matrix \(A\).

In the context of quadratic forms, symmetric matrices help us structure expressions neatly. Given a quadratic form like \(q = 3u^2 - 4uv + 7v^2\), it can be expressed as \(\mathbf{x}^T \mathbf{A} \mathbf{x}\), where \(\mathbf{x}\) is a vector of variables, and \(\mathbf{A}\) is a symmetric matrix. This ensures efficient calculations and simple verification, as the matrix encapsulates all coefficient information in a structured manner.

Symmetric matrices mean that the order of multiplication doesn't impact the end result. This property aids in rearranging expressions without introducing errors.

Matrix Product

A matrix product involves multiplying matrices in a manner that aligns rows and columns correctly for result derivation. This is crucial when expressing quadratic forms using matrices, to ensure accurate representations of relationships among variables. In mathematical terms, if \( \mathbf{A} \) and \( \mathbf{B} \) are matrices, their product \( \mathbf{A} \mathbf{B} \) will be a new matrix \( \mathbf{C} \).

For quadratic forms, we look at the product result of \( \mathbf{x}^T \mathbf{A} \mathbf{x} \). Here, the vector \( \mathbf{x} \) contains all variables involved (e.g., \(u, v\)), \( \mathbf{x}^T \) is its transpose, and \( \mathbf{A} \) is the symmetric matrix holding the coefficients. Each element in the resulting matrix product represents a component of the original quadratic expression.

This arrangement makes verifying and understanding complex mathematical relationships far simpler.

Coefficient Matrix

The coefficient matrix is central to expressing quadratic forms as matrix products. It includes all coefficients from the quadratic expression, neatly organized in a matrix form. This matrix is always symmetric for quadratic forms.

For example, consider the quadratic form \(q=u^2+7uv+3v^2\). Its coefficient matrix is determined as follows:

• The diagonal elements represent coefficients of individual variables squared (here, 1 for \(u^2\) and 3 for \(v^2\)).
• The off-diagonal elements are halved coefficients of the cross product terms (3.5 for \(uv\), reflecting the original full coefficient of 7 when considering symmetry).

This setup helps us quickly interpret both individual and interaction effects among variables, unifying expression into a consistent matrix framework.

Cross Product Terms

Cross product terms in quadratic forms describe how different variables interact in the equation. These terms generally appear with coefficients and are key to forming the off-diagonal components of the symmetric coefficient matrix.

In terms like \(uv, xy, \) these indicate interaction between variable pairs, which mathematicians represent in matrices' off-diagonal entries. For example, in the expression \(q = 4uv\), the cross product term is \(4uv\) and contributes to both the \(u,v\) and \(v,u\) positions in the symmetric matrix. However, to maintain symmetry, you take half the coefficient value.

Understanding cross product terms' role is essential for constructing the entire mathematical vehicle that expresses complex relationships in a simplified, coherent matrix form. This way, they serve as a bridge connecting raw mathematical relationships to matrix representation, explaining their interaction terms effectively.