Rearrange This Expression Into Quadratic Form

Unlocking the Power of Quadratic Form: A Comprehensive Guide

Quadratic forms are fundamental in mathematics, appearing in various fields like linear algebra, optimization, and number theory. Mastering the art of rearranging expressions into quadratic form unlocks powerful tools for analysis and problem-solving. This guide provides a comprehensive understanding of quadratic forms, focusing on the techniques needed to manipulate expressions into this essential form.

What is a Quadratic Form?

A quadratic form is a homogeneous polynomial of degree two in n variables. In simpler terms, it's an expression where each term has a total degree of two. This means each term can be:

• A squared variable (e.g., x², y²).
• A product of two different variables (e.g., xy, yz, xz).

The general form of a quadratic form in n variables, x₁, x₂, ..., x_n, can be written as:

Q(x₁, x₂, ..., x_n) = ∑_i=1ⁿ ∑_j=1ⁿ a_ijx_ix_j

Where a_ij are coefficients. This summation represents all possible combinations of variables, each multiplied by a constant coefficient.

Representing Quadratic Forms with Matrices:

Quadratic forms are elegantly represented using matrices. Any quadratic form can be written as:

Q(x) = x^TAx

Where:

• x is a column vector of variables (x₁, x₂, ..., x_n)^T.
• x^T is the transpose of the vector x.
• A is a symmetric n x n matrix of coefficients. Symmetry implies that a_ij = a_ji. This is crucial because it ensures the uniqueness of the matrix representation.

Why Symmetric Matrices?

Using a symmetric matrix is a standard convention that simplifies analysis. For any quadratic form, there exists a unique symmetric matrix representation. This avoids ambiguity and facilitates techniques like diagonalization to analyze the form. If A is not symmetric, it can be symmetrized by replacing A with (A + A^T)/2, as x^TAx = x^T((A + A^T)/2)x when x is a vector of real variables.

Identifying Quadratic Forms

Before rearranging an expression, you must first determine if it can be expressed as a quadratic form. Look for these characteristics:

• Degree Two: All terms must have a total degree of two. Linear terms (e.g., x, y) or constant terms (e.g., 5, -2) are not allowed in a pure quadratic form.
• Homogeneity: The expression must be homogeneous, meaning all terms have the same degree.
• Presence of Squared Terms and Cross-Product Terms: Look for terms like x², y², z², xy, xz, yz, etc.

Examples of Quadratic Forms:

• x² + 2xy + y²
• 3x² - 5yz + z²
• x² (This is a quadratic form in one variable.)

Examples of Expressions That Are NOT Quadratic Forms:

• x + y (Linear terms present)
• x² + y + 1 (Linear and constant terms present)
• x³ + y² (Terms with different degrees)

Rearranging Expressions into Quadratic Form: The Process

The core task is to manipulate a given expression to match the structure of x^TAx. Here's a step-by-step process:

1. Identify Variables: Determine the variables present in the expression (e.g., x, y, z).

2. Collect Squared Terms: Group all terms containing squared variables. These will correspond to the diagonal elements of the matrix A.

3. Collect Cross-Product Terms: Group all terms containing products of two different variables. These will correspond to the off-diagonal elements of the matrix A. Remember that a_ij = a_ji in the symmetric matrix. Therefore, the coefficient of x_ix_j needs to be split equally between a_ij and a_ji.

4. Construct the Matrix A: Create the symmetric matrix A based on the coefficients identified in steps 2 and 3.

• Diagonal Elements: The coefficients of the squared terms (x₁², x₂², ..., x_n²) become the diagonal elements of A (a₁₁, a₂₂, ..., a_nn).

• Off-Diagonal Elements: The coefficients of the cross-product terms (x_ix_j) are divided by 2 and placed in the corresponding off-diagonal positions (a_ij and a_ji).

5. Form the Vector x: Create the column vector x containing the variables in a consistent order (e.g., x = (x, y, z)^T).

6. Write the Quadratic Form: Express the rearranged expression as x^TAx.

Examples with Detailed Explanations

Let's illustrate the process with several examples.

Example 1: Two Variables

Expression: Q(x, y) = 2x² + 6xy + y²

1. Variables: x, y

2. Squared Terms: 2x², y²

3. Cross-Product Terms: 6xy

4. Construct Matrix A:

   • a₁₁ = 2 (coefficient of x²)
   • a₂₂ = 1 (coefficient of y²)
   • a₁₂ = a₂₁ = 6/2 = 3 (half the coefficient of xy)

   Therefore, A = [[2, 3], [3, 1]]

5. Form Vector x: x = [x, y]^T

6. Write Quadratic Form: Q(x, y) = [x, y] [[2, 3], [3, 1]] [x, y]^T

Example 2: Three Variables

Expression: Q(x, y, z) = x² - 4xy + 2yz + 5z²

1. Variables: x, y, z

2. Squared Terms: x², 5z² (Notice that y² is missing, which is perfectly acceptable; it simply means its coefficient is zero.)

3. Cross-Product Terms: -4xy, 2yz

4. Construct Matrix A:

   • a₁₁ = 1 (coefficient of x²)
   • a₂₂ = 0 (coefficient of y², which is missing)
   • a₃₃ = 5 (coefficient of z²)
   • a₁₂ = a₂₁ = -4/2 = -2 (half the coefficient of xy)
   • a₂₃ = a₃₂ = 2/2 = 1 (half the coefficient of yz)
   • a₁₃ = a₃₁ = 0 (coefficient of xz is missing)

   Therefore, A = [[1, -2, 0], [-2, 0, 1], [0, 1, 5]]

5. Form Vector x: x = [x, y, z]^T

6. Write Quadratic Form: Q(x, y, z) = [x, y, z] [[1, -2, 0], [-2, 0, 1], [0, 1, 5]] [x, y, z]^T

Example 3: A More Complex Case

Expression: Q(x, y, z) = 4x² + y² - 9z² + 8xy - 4xz + 6yz

1. Variables: x, y, z

2. Squared Terms: 4x², y², -9z²

3. Cross-Product Terms: 8xy, -4xz, 6yz

4. Construct Matrix A:

   • a₁₁ = 4 (coefficient of x²)
   • a₂₂ = 1 (coefficient of y²)
   • a₃₃ = -9 (coefficient of z²)
   • a₁₂ = a₂₁ = 8/2 = 4 (half the coefficient of xy)
   • a₁₃ = a₃₁ = -4/2 = -2 (half the coefficient of xz)
   • a₂₃ = a₃₂ = 6/2 = 3 (half the coefficient of yz)

   Therefore, A = [[4, 4, -2], [4, 1, 3], [-2, 3, -9]]

5. Form Vector x: x = [x, y, z]^T

6. Write Quadratic Form: Q(x, y, z) = [x, y, z] [[4, 4, -2], [4, 1, 3], [-2, 3, -9]] [x, y, z]^T

Completing the Square and Quadratic Forms

Completing the square is a powerful technique closely related to quadratic forms. It allows you to rewrite a quadratic expression in a form that reveals its vertex or minimum/maximum value. While not directly rearranging into the matrix form x^TAx, it's a useful skill for analyzing quadratic expressions and understanding their properties.

The Basic Idea:

The goal of completing the square is to rewrite a quadratic expression of the form ax² + bx + c into the form a( x + h )² + k. The values of h and k provide information about the vertex of the parabola represented by the quadratic expression.

Steps for Completing the Square:

1. Factor out 'a': If a ≠ 1, factor out a from the x² and x terms: a(x² + ( b/a )x) + c

2. Find the Value to Complete the Square: Take half of the coefficient of the x term inside the parentheses (b/a), square it, and add and subtract it inside the parentheses: a(x² + ( b/a )x + ( b/2a )² - ( b/2a )²) + c

3. Rewrite as a Squared Term: The first three terms inside the parentheses now form a perfect square: a[( x + ( b/2a ) )² - ( b/2a )²] + c

4. Simplify: Distribute the a and combine the constant terms: a( x + ( b/2a ) )² + c - a( b/2a )²

Example:

Complete the square for the expression: x² + 6x + 5

1. a = 1, so no factoring is needed.

2. Half of 6 is 3, and 3² = 9. Add and subtract 9: x² + 6x + 9 - 9 + 5

3. Rewrite as a squared term: ( x + 3 )² - 9 + 5

4. Simplify: ( x + 3 )² - 4

Therefore, x² + 6x + 5 = ( x + 3 )² - 4. The vertex of the parabola is at (-3, -4).

Connecting Completing the Square to Quadratic Forms:

While completing the square doesn't directly create the x^TAx form, it can be used to simplify quadratic expressions before converting them into matrix form. It's especially useful when dealing with quadratic expressions that also contain linear terms. By completing the square, you can isolate the purely quadratic part of the expression and then rearrange it into matrix form.

Applications of Quadratic Forms

Quadratic forms have wide-ranging applications across various fields:

• Optimization: Determining the minimum or maximum values of functions, especially in constrained optimization problems. The definiteness of the matrix A (positive definite, negative definite, etc.) determines the nature of the extremum.

• Conic Sections: Classifying conic sections (ellipses, hyperbolas, parabolas) based on the coefficients of the quadratic form representing the conic.

• Statistics: Representing covariance matrices and Mahalanobis distance, used for measuring the distance between a point and a distribution.

• Physics: Describing energy functions in classical mechanics and quantum mechanics.

• Machine Learning: Feature extraction, dimensionality reduction (e.g., Principal Component Analysis), and kernel methods.

Common Mistakes to Avoid

• Forgetting Symmetry: Ensure the matrix A is symmetric. Always split the coefficient of x_ix_j equally between a_ij and a_ji.

• Incorrectly Handling Missing Terms: If a squared term or cross-product term is missing, its corresponding coefficient in the matrix A is zero. Don't leave those entries blank.

• Mixing Up Variables: Be consistent with the order of variables in the vector x. The order must match the rows and columns of the matrix A.

• Ignoring Linear or Constant Terms: Remember that the pure quadratic form only includes terms of degree two. Linear and constant terms must be treated separately or removed if you need to represent only the quadratic part.

Advanced Topics and Further Exploration

This guide provides a solid foundation in rearranging expressions into quadratic form. To deepen your understanding, explore these advanced topics:

• Diagonalization of Quadratic Forms: Finding a change of variables that transforms the quadratic form into a sum of squares. This simplifies analysis and reveals important properties.

• Sylvester's Law of Inertia: This law states that the number of positive, negative, and zero eigenvalues of the matrix A is invariant under congruence transformations. This is related to the definiteness of the quadratic form.

• Definiteness of Quadratic Forms: Classifying quadratic forms as positive definite, negative definite, positive semi-definite, negative semi-definite, or indefinite based on the eigenvalues of the matrix A. This determines the nature of the function's extremum.

• Applications in Constrained Optimization: Using Lagrange multipliers to solve optimization problems involving quadratic forms subject to constraints.

Conclusion

Rearranging expressions into quadratic form is a valuable skill with broad applications. By understanding the fundamental concepts and mastering the step-by-step process, you can unlock the power of quadratic forms for analysis, problem-solving, and deeper insights across various mathematical and scientific disciplines. Remember to focus on the symmetry of the matrix representation and practice regularly to solidify your understanding. As you delve deeper into related topics like diagonalization and definiteness, you'll discover the remarkable versatility and elegance of quadratic forms.