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Chapter 09: Quadratic and Bilinear Forms

Prerequisites: Matrix Algebra (Ch02) · Eigenvalues and Diagonalization (Ch06) · Vector Spaces (Ch04)

Chapter Outline: Definition of Bilinear Forms → Symmetric Bilinear Forms → Definition of Quadratic Forms \(\mathbf{x}^T A \mathbf{x}\) → Canonical and Normal Forms → Completing the Square and Elementary Transformations → Congruence Transformations → Sylvester's Law of Inertia (Inertia Indices) → Definiteness Criteria (Hurwitz Criterion) → Geometric Interpretation: Classification and Rotation of Quadric Surfaces → Applications: Extrema Analysis of Multivariate Functions (Hessian Matrices)

Extension: Quadratic forms are the algebraic characterization of local shapes of multivariate functions; they are the algebraic core of kinetic energy expressions in classical mechanics and optimization theory; they provide the foundation for studying manifold curvature and gravitational field equations.

Quadratic forms study scalar functions that take the form of homogeneous polynomials of degree two. By linear substitution (congruence transformations), we can eliminate complex cross-product terms, thereby revealing the essential geometric features of the function. This chapter establishes standard methods for determining the "sign" of a quadratic form and maps algebraic properties perfectly to the morphology of spatial surfaces.

9.1 Bilinear and Quadratic Forms

Definition 09.1 (Bilinear and Quadratic Forms)

  1. Bilinear Form: A mapping \(B: V \times V \to F\) that is linear in each argument separately.
  2. Quadratic Form: A function \(Q(\mathbf{x}) = B(\mathbf{x}, \mathbf{x})\) derived from a symmetric bilinear form. In a given basis, it can be written as: $\(Q(\mathbf{x}) = \sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j = \mathbf{x}^T A \mathbf{x}\)$ where \(A\) can always be chosen as a symmetric matrix.

9.2 Congruence and Canonical Forms

Definition 09.2 (Congruence)

Two matrices \(A\) and \(B\) are congruent if there exists a non-singular matrix \(P\) such that \(B = P^T A P\). Geometric Insight: Congruence corresponds to a linear change of coordinates and does not change the set of values attained by the quadratic form.

Theorem 09.1 (Sylvester's Law of Inertia)

When a quadratic form over \(\mathbb{R}\) is reduced to canonical form \(\sum d_i y_i^2\) via congruence, the number of positive coefficients \(p\) (positive inertia index) and negative coefficients \(q\) (negative inertia index) are invariants. - Rank: \(r = p + q\). - Signature: \(s = p - q\).

9.3 Criteria for Definiteness

Definition 09.3 (Definiteness Classification)

  1. Positive Definite: \(Q(\mathbf{x}) > 0\) for all \(\mathbf{x} \neq \mathbf{0}\). Eigenvalues are all positive.
  2. Positive Semi-definite: \(Q(\mathbf{x}) \ge 0\) for all \(\mathbf{x}\). Eigenvalues are non-negative.
  3. Indefinite: Takes both positive and negative values. Eigenvalues have mixed signs.

Theorem 09.2 (Hurwitz Criterion / Leading Principal Minors)

A real symmetric matrix \(A\) is positive definite iff all its leading principal minors are strictly positive.

Exercises

1. [Basics] Write the symmetric matrix representation for \(f(x, y) = x^2 + 4xy + 3y^2\).

Solution

Steps: 1. Place squared coefficients on the diagonal: \(a_{11}=1, a_{22}=3\). 2. Split the cross-term coefficient 4 equally: \(a_{12}=2, a_{21}=2\). Matrix Representation: \(A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}\). Verification: \((x \ y) \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = x(x+2y) + y(2x+3y) = x^2 + 4xy + 3y^2\).

2. [Completion] Use the method of completing the square to find the canonical form of \(Q(x, y) = x^2 + 4xy\).

Solution

Steps: 1. Complete the square: \(Q = (x^2 + 4xy + 4y^2) - 4y^2\). 2. Write as a full square: \(Q = (x+2y)^2 - 4y^2\). 3. Let \(u = x+2y, v = y\). Conclusion: The canonical form is \(u^2 - 4v^2\). Its inertia indices are \(p=1, q=1\).

3. [Congruence] If \(A\) and \(B\) are congruent, must they have the same eigenvalues?

Solution

Conclusion: Not necessarily. Analysis: - Similarity \(P^{-1}AP\) preserves eigenvalues. - Congruence \(P^T AP\) preserves inertia indices (the number of positive/negative eigenvalues) but not the specific values. - For example, \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\) are congruent (let \(P=\sqrt{2}I\)), but have different eigenvalues.

4. [Definiteness] Use principal minors to determine if \(A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}\) is positive definite.

Solution

Calculation: 1. 1st minor: \(D_1 = 2 > 0\). 2. 2nd minor: \(D_2 = 2 \cdot 2 - 1 \cdot 1 = 3 > 0\). Conclusion: Since all leading principal minors are positive, the matrix is positive definite.

5. [Inertia] Find the inertia and signature of \(Q = x^2 - y^2\).

Solution

Analysis: 1. The form is already canonical: \(1x^2 + (-1)y^2\). 2. Number of positive coefficients \(p=1\). 3. Number of negative coefficients \(q=1\). 4. Signature \(s = p - q = 0\). Conclusion: The inertia is \((1, 1)\), and the signature is 0.

6. [Geometry] What curves do \(x^2 + y^2 = 1\) and \(x^2 - y^2 = 1\) represent?

Solution

Classification: 1. \(x^2 + y^2 = 1\): Represents a circle (or ellipse). The corresponding quadratic form is positive definite, resulting in closed level sets. 2. \(x^2 - y^2 = 1\): Represents a hyperbola. The corresponding quadratic form is indefinite, resulting in open level sets. This shows how the sign character of a quadratic form dictates the topology of its geometric level sets.

7. [Rayleigh] What is the maximum value of the Rayleigh Quotient \(R(\mathbf{x}) = \frac{\mathbf{x}^T A \mathbf{x}}{\mathbf{x}^T \mathbf{x}}\)?

Solution

Theorem: The maximum value of the Rayleigh Quotient is the largest eigenvalue \(\lambda_{\max}\) of \(A\), and the minimum is \(\lambda_{\min}\). This builds a bridge between geometric "stretching" and the algebraic spectrum.

8. [Skew-symmetry] Prove that for any skew-symmetric matrix \(A\), the induced quadratic form \(\mathbf{x}^T A \mathbf{x}\) is identically 0.

Solution

Proof: 1. A quadratic form is a scalar, so it equals its transpose: \(\mathbf{x}^T A \mathbf{x} = (\mathbf{x}^T A \mathbf{x})^T\). 2. Expand: \(= \mathbf{x}^T A^T \mathbf{x}\). 3. Use \(A^T = -A\): \(= \mathbf{x}^T (-A) \mathbf{x} = -\mathbf{x}^T A \mathbf{x}\). 4. A number equal to its negative must be 0. Conclusion: This is why we only consider symmetric matrices when studying quadratic forms.

9. [Normal Form] Transform \(Q = 2x^2 + 2y^2\) into normal form.

Solution

Steps: 1. It's already in canonical form. 2. To get the normal form (coefficients in \(\{1, -1, 0\}\)), scale the variables. 3. Let \(u = \sqrt{2}x, v = \sqrt{2}y\). Conclusion: The normal form is \(u^2 + v^2\).

10. [Application] How is a quadratic form used to determine the nature of critical points in optimization?

Solution

Analysis: Using the Hessian matrix \(H\) of a multivariate function: 1. If \(H\) at a critical point is positive definite, the point is a local minimum. 2. If \(H\) is negative definite, it is a local maximum. 3. If \(H\) is indefinite, it is a saddle point. This illustrates the core role of quadratic forms in analyzing the curvature of continuous space.

Chapter Summary

Quadratic forms bridge polynomial algebra and spatial geometry:

  1. Structural Simplification: Congruence transformations are the blades that eliminate cross-terms and simplify coordinate systems, revealing the core inertia of symmetric operators.
  2. Sign Dynamics: The Law of Inertia establishes the deepest topological invariants of a quadratic function, which remain unchanged regardless of coordinate rotation or scaling.
  3. Core of Definiteness: As the algebraic criterion for energy, distance, and stability analysis, the theory of definiteness is the vital junction connecting linear algebra with analysis and physics.