1 Definition and Basic Examples

A quadratic form is a homogeneous polynomial of degree two in one or more variables. In linear algebra, it is usually viewed as a function that takes a vector as input and returns a scalar. The value depends on how the vector’s components are combined, and the expression is governed by second-degree terms only. Quadratic forms appear naturally in geometry, algebra, and optimization because they measure curved behavior and capture invariant structure.

1.1 Quadratic forms in coordinates

In coordinates, a quadratic form in variables \(x_1,\dots,x_n\) is written as a sum of terms of the form \(a_{ij}x_ix_j\). Only degree-two terms occur, so there are no linear or constant terms. For example, expressions such as \(x^2+3xy+2y^2\) and \(2u^2-v^2\) are quadratic forms.

The coordinate expression makes it easy to study how the form behaves under substitution and change of basis. It also shows how cross terms such as \(xy\) contribute to the overall shape of the form.

1.2 Matrix representation \(q(x)=x^\top A x\)

A quadratic form is often represented using a matrix \(A\) by \[ q(x)=x^\top A x, \] where \(x\) is a column vector and \(x^\top\) is its transpose. When \(A\) is symmetric, this representation is especially convenient, since the quadratic form is then determined completely by the entries of \(A\).

In this setting, diagonal entries contribute squared terms, while off-diagonal entries produce mixed terms. For instance, a term \(xy\) may arise from symmetric entries in two positions of the matrix.

1.3 Homogeneity and symmetry requirements

Quadratic forms are homogeneous of degree two, meaning that scaling the vector by a factor \(t\) scales the value by \(t^2\). This property distinguishes them from more general polynomial functions. Symmetry is also important: when a matrix representation is used, the form is typically associated with a symmetric matrix, because antisymmetric parts do not affect \(x^\top A x\).

This symmetry is one reason quadratic forms connect closely with bilinear forms and orthogonal geometry.

1.4 Common low-dimensional examples (1D and 2D)

In one dimension, a quadratic form has the simple shape \(q(x)=ax^2\). Its behavior depends entirely on the sign of \(a\). If \(a>0\), the form is always nonnegative except at zero; if \(a<0\), it is always nonpositive.

In two dimensions, examples include \(x^2+y^2\), \(x^2-y^2\), and \(xy\). These illustrate different geometric behaviors: circular symmetry, saddle-like behavior, and mixed interaction between coordinates. Even in low dimensions, quadratic forms already show the main patterns found in higher-dimensional cases.

2 Algebraic Properties

Quadratic forms have a rich algebraic structure. They can be expanded, rewritten in terms of bilinear expressions, and transformed by linear maps. These properties make them useful for classification and computation.

2.1 Expansion rules and bilinear interpretation

When expanded, a quadratic form separates into square terms and cross terms. The coefficient of a mixed term is often split between two symmetric positions in the matrix representation. This convention ensures that the form can be encoded by a bilinear rule.

For many purposes, it is helpful to view the quadratic form as arising from a map that takes two vectors and produces a scalar, then specialize the two vectors to be equal. This viewpoint clarifies how the coefficients combine under algebraic manipulation.

2.2 Relation to bilinear forms

Every symmetric bilinear form gives rise to a quadratic form by evaluating both arguments on the same vector. Conversely, over many standard settings, a quadratic form can be recovered from an associated symmetric bilinear form. This relationship explains why quadratic forms are often studied alongside bilinear forms.

The connection is especially important in linear algebra, where bilinear forms provide a more flexible framework for proving identities and deriving invariants.

2.3 Equivalent matrix representations

The same quadratic form can be represented by more than one matrix if the matrix is not assumed to be symmetric. However, all matrices that differ by an antisymmetric part define the same quadratic expression \(x^\top A x\). For this reason, one often chooses a symmetric representative.

Equivalent representations are useful because they allow the form to be simplified while preserving its essential algebraic content.

2.4 Scaling, addition, and linear transformations

If a quadratic form is multiplied by a scalar, its coefficients are scaled accordingly. Two quadratic forms can also be added term by term to produce a new one. These operations are straightforward algebraically and often appear in decomposition arguments.

Under a linear change of variables, the form transforms in a structured way that preserves its degree and general type. This makes quadratic forms stable objects for study under coordinate changes.

3 Diagonalization and Normal Forms

A major theme in the theory of quadratic forms is simplification through change of basis. By choosing coordinates carefully, one can often eliminate cross terms and obtain a diagonal or otherwise standardized expression.

3.1 Change of variables and congruence

Changing variables in a quadratic form corresponds to replacing the vector \(x\) by a linear transformation of new coordinates. On the matrix side, this leads to congruence transformations rather than similarity transformations. The new matrix remains equivalent to the original form, but its entries may be much simpler.

This process preserves the quadratic nature of the expression while revealing its essential structure.

3.2 Diagonalization over the real numbers

Over the real numbers, many quadratic forms can be diagonalized by an appropriate linear change of basis. The resulting expression contains only squared terms, with no mixed products. This is one of the most useful simplifications in the subject.

Diagonalization makes it easier to determine whether a form is positive, negative, or indefinite, and it often reduces the study of a form to a list of scalar coefficients.

3.3 Sylvester’s criterion and canonical structure

Sylvester’s criterion gives a practical method for testing whether a symmetric matrix is positive definite using leading principal minors. More broadly, real quadratic forms admit canonical structures that classify them up to change of basis. In such classifications, the form is reduced to a standard arrangement of \(+1\), \(-1\), and \(0\) terms.

These canonical descriptions capture the core behavior of the form and are central to classification theory.

3.4 Rank and the reduced form

The rank of a quadratic form is the rank of its associated matrix and measures how many independent directions contribute nontrivially. A reduced form is one in which zero directions and dependent components have been removed or isolated.

Rank helps distinguish degenerate forms from nondegenerate ones. It also indicates how many variables actually affect the value of the form.

4 Spectral and Geometric Interpretation

Quadratic forms have a strong geometric meaning. Their matrices determine directions of stretching and compression, and their level sets describe familiar surfaces and curves.

4.1 Eigenvalues and eigenvectors of the associated matrix

When the associated matrix is symmetric, it has real eigenvalues and orthogonal eigenvectors. In the eigenvector basis, the quadratic form becomes diagonal, with each eigenvalue serving as a coefficient of a squared coordinate. Positive and negative eigenvalues contribute with opposite signs, while zero eigenvalues indicate flat directions.

This spectral viewpoint is one of the clearest ways to understand the form’s geometry.

4.2 Oriented geometry: ellipsoids, hyperboloids, and cones

The equation \(q(x)=c\) can define surfaces such as ellipsoids, hyperboloids, or cones depending on the signs and values of the coefficients. A positive definite form typically yields ellipsoidal level sets, while mixed signs lead to saddle-like or hyperbolic shapes. When the form is degenerate, the level set may collapse into a cone or pair of planes.

These shapes show how algebraic data translates directly into spatial structure.

4.3 Kernel and null directions

The kernel of the associated matrix consists of vectors \(x\) for which \(q(x)\) vanishes in directions that do not contribute to the form. Such null directions are important in understanding degeneracy. If the kernel is nontrivial, the form does not respond to movement along some subspace.

Kernel analysis helps identify flat directions and determine whether the form is fully nondegenerate.

4.4 Level sets and classification intuition

Level sets of a quadratic form are the collections of vectors that produce the same scalar value. Studying these sets gives intuition about the overall classification of the form. Shapes that are closed and bounded indicate one kind of behavior, while open or singular surfaces indicate another.

This geometric perspective is often the fastest route to understanding how the signs of the coefficients affect the form.

5 Definiteness and Classification

Quadratic forms are commonly classified by the sign of their values. Definiteness is a central concept because it determines whether the form acts like a squared length, a reversed squared length, or a saddle.

5.1 Positive definite, negative definite, and indefinite

A quadratic form is positive definite if it is strictly positive for every nonzero vector. It is negative definite if it is strictly negative for every nonzero vector. A form is indefinite if it takes both positive and negative values on different vectors.

These categories are fundamental and often determine the role of the form in geometry and analysis.

5.2 Semidefinite cases and boundary behavior

A semidefinite form is nonnegative or nonpositive but not strictly so. In this case, the form can vanish on nonzero vectors, producing boundary behavior between definite and indefinite cases. Such forms often arise when one or more eigenvalues are zero.

Semidefiniteness is important in settings where a quantity measures energy, distance, or curvature but may flatten in certain directions.

5.3 Signature (in real settings)

The signature of a real quadratic form records the numbers of positive and negative terms in a diagonalized representation. Zero terms are often tracked separately when the form is degenerate. The signature provides a compact invariant that summarizes the form’s basic sign structure.

Because it remains unchanged under appropriate coordinate changes, the signature is a powerful classification tool.

5.4 Connections to optimization and constraint curvature

In optimization, quadratic forms often describe curvature near a critical point. A positive definite form indicates local convexity, while an indefinite form suggests saddle behavior. In constrained problems, the form may arise as part of a second-order test.

This connection makes quadratic forms useful not only in algebra but also in the analysis of extremal behavior.

6 Transformations and Congruence Invariants

The most meaningful properties of a quadratic form are those that survive change of basis. These invariants are central to classification and comparison.

6.1 Congruence vs similarity

Quadratic forms transform by congruence, meaning the matrix changes as \(A \mapsto P^\top A P\) under a basis change. This differs from similarity, which uses \(P^{-1}AP\). The distinction matters because congruence preserves the quadratic expression, while similarity preserves linear operators.

Understanding this difference is essential for working correctly with forms and matrices.

6.2 Invariants under change of basis

Certain quantities remain unchanged under congruence, including rank and signature in the real symmetric case. These invariants determine whether two quadratic forms are equivalent under a linear coordinate transformation. They therefore provide a compact way to compare forms.

Because of these invariants, classification can often be reduced to a finite list of standard models.

The determinant of the associated matrix gives information about nondegeneracy: a nonzero determinant indicates full rank. The trace, while not invariant under congruence in general, can still be useful in specific coordinate descriptions or after diagonalization. Rank remains one of the most reliable descriptors, since it directly measures the number of effective variables.

Together, these quantities help summarize the size and structure of the form.

6.4 Behavior under scaling of the form

Multiplying a quadratic form by a nonzero scalar changes its numerical values but usually preserves its essential shape up to sign and scale. Positive scaling does not alter definiteness, while negative scaling swaps positive and negative behavior. Zero scaling collapses the form completely.

This simple operation illustrates how quadratic forms can be compared by their qualitative structure rather than by exact coefficients.

7 Applications and Further Topics (Algebraic)

Quadratic forms appear throughout algebra and classical geometry. They provide a common language for equations, factorization methods, and field-dependent classification.

7.1 Systems of quadratic equations

Collections of quadratic equations can often be studied by isolating the quadratic part of each expression. Even when the full system is more complicated, the quadratic form controls the leading behavior. In many problems, the geometry of intersections is determined first by these second-degree terms.

This makes quadratic forms a foundational tool in the study of polynomial systems.

7.2 Quadratic forms over different fields

The theory of quadratic forms varies depending on the underlying field. Over the real numbers, sign and signature play a central role. Over other fields, such as the rational numbers or finite fields, different invariants become important, and classification may be more subtle.

Field dependence is one reason the subject has both algebraic depth and broad applicability.

7.3 Bilinear factorization and completing the square

Completing the square is a standard method for rewriting a quadratic form into a simpler expression. In two variables, it can remove mixed terms or isolate a square plus a remainder. More generally, factorization ideas and bilinear identities help transform a complicated form into a more manageable one.

These techniques are especially useful in solving equations and deriving normal forms.

7.4 Connections to conic sections and classical geometry

In two variables, quadratic forms describe conic sections after suitable coordinate choices. Circles, ellipses, parabolas, and hyperbolas all arise from second-degree equations, with the quadratic part governing the basic shape. Classical geometry often interprets these curves through their associated matrices and invariants.

This geometric role has made quadratic forms a lasting part of the language of mathematics.