Quadratic forms, Equivalence, Reduction to canonical form, Lagrange's Reduction, Sylvester's law of inertia, Definite and semi-definite forms, Regular quadratic form

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Quadratic forms, Equivalence, Reduction to canonical form, Lagrange’s Reduction, Sylvester’s law of inertia, Definite and semi-definite forms, Regular quadratic form

Quadratic form. A quadratic form is a homogeneous polynomial of degree two. The following are quadratic forms in one, two and three variables:

F(x) = ax²

F(x,y) = ax² + by² + cxy

F(x,y,z) = ax² + by² + cz² + dxy + exz + fyz

The polynomial consists of squared terms for each of the variables plus cross-products terms for all combinations of the variables.

Quadratic forms occur in many branches of mathematics and its applications. They are encountered in the theory of numbers, in crystallography, in the study of surfaces in analytic geometry, and in various problems of physics and mechanics.

It is customary to represent the quadratic form as a symmetric bilinear form

= X^TAX

where A is a symmetric matrix of the coefficients. Note that in this representation each cross-product term appears twice. For example, the x₁x₂ term appears as both an x₁x₂ term and a x₂x₁ term. Because we want the matrix to be symmetric we allocate half of the value of the coefficient to each term. The motivation behind representing the quadratic form in this manner is to facilitate matrix analysis and treatment.

The matrix A is called the matrix of the quadratic form and the rank of A is called the rank of the form. If the rank is less than n the quadratic form is called singular. If the rank is equal to n it is nonsingular.

Change of variables. Let us consider the effect of a change of variables on the form. Let X = BY. Then

X^TAX = (BY)^TA(BY) = Y^T(B^TAB)Y

Thus the linear transformation X = BY carries the quadratic form X^TAX with a symmetric matrix A into the quadratic form Y^T(B^TAB)Y with symmetric matrix B^TAB. Matrix B may be either singular or nonsingular. If B is nonsingular matrix A is congruent to matrix B^TAB.

Equivalence of quadratic forms. Two quadratic forms are said to be equivalent over a field F if and only if there exists a nonsingular linear transformation X = BY that carries one of the forms into the other.

Two quadratic forms over a field F are equivalent over F if and only if their matrices are congruent over F.

Reduction to canonical form.

Any quadratic form over a field F of rank r can be reduced to the form

(1)

by a non-singular linear transformation X = BY.

Connection to quadric surfaces of solid analytic geometry. Let X^TAX be the quadratic form associated with some quadric surface in three-dimensional space. Let us say it is some ellipsoid. in 3-space. Let Y = PX represent that change of variables that reduces the quadratic form to

where P is orthogonal. Then this change of variables Y = BX represents that change of basis that reduces the expression of the quadric surface to its canonical form. The numbers k₁, k₂, .... ,k_r are the eigenvalues of A. The eigenvectors of matrix A correspond to directions of the principal axes of the quadric surface.

Lagrange’s Reduction. The reduction of a quadratic form to the form (1) above can be carried out by a procedure known as Lagrange’s Reduction, which consists essentially of repeated completing of the square.

Example.

Inspection of this last expression for q shows those substitutions that will reduce q to the canonical form of (1) above.

Let

Substituting y₁, y₂ and y₃ into the last expression for q gives

which is of the canonical form (1) above where q is expressed in terms of the new variables y₁, y₂ and y₃.

Real quadratic forms.

Theorem. Any quadratic form over the field of real numbers can be reduced by a non-singular linear transformation to the canonical form

(2)

where the number p of positive terms is called the index and r is the rank of the quadratic form. The number of positive terms minus the number of negative terms, p - (r - p), is called the signature of the quadratic form.

Index and signature of symmetric and Hermitian matrices. The index of a symmetric or of a Hermitian matrix is the number of positive elements when it is transformed to diagonal form. The number of positive elements minus the number of negative elements is the signature and the total number of nonzero elements is the rank.

Sylvester’s law of inertia. Let a real quadratic form be reduced by two different real non-singular transformations to canonical forms of type (2). Then the two canonical forms to which it is reduced have the same rank and the same index.

Thus the number p of positive terms which appear in the reduced form (2) is an invariant of the given form, in the sense that p depends only on the form and not on the method used to reduce it. In other words, two quadratic forms (or two symmetric or Hermitian matrices) have the same rank and the same index if and only if they are congruent, i.e. if one can be transformed into the other by an invertible linear transformation.

Two real quadratic forms each in n variables are equivalent over the real field if and only if they have the same rank and the same index or the same rank and the same signature.

Complex quadratic forms.

Every quadratic form over the complex field of rank r can be reduced by a nonsingular transformation over the complex field to the canonical form

Two complex quadratic forms each in n variables are equivalent over the complex field if and only if they have the same rank.

Definite and semi-definite forms.

Positive definite quadratic form. A real non-singular quadratic form q = X^TAX in n variables is called positive definite if its rank and index are equal. Because it is nonsingular its rank r is equal to n. Thus, in the real field, a positive definite quadratic form can be reduced to y₁² + y₂² + ... + y_n². Its value q is greater than zero for any non-trivial value of X i.e. any value of X with the exception of the trivial case

Positive semi-definite quadratic form. A real singular quadratic form q = X^TAX in n variables is called positive semi-definite if its rank and index are equal. Because it is singular its rank r is less than n. Thus, in the real field, a positive semi-definite quadratic form can be reduced to y₁² + y₂² + ... + y_r² where r < n.. Its value q is greater than or equal to zero for any non-trivial value of X.

Negative definite quadratic form. A real non-singular quadratic form q = X^TAX in n variables is called negative definite if its index p = 0. Because it is nonsingular its rank r is equal to n. Thus, in the real field, a negative definite quadratic form can be reduced to -y₁² - y₂² - ... - y_n². Its value q is less than zero for any non-trivial value of X.

Negative semi-definite quadratic form. A real singular quadratic form q = X^TAX in n variables is called negative semi-definite if its index p = 0. Because it is singular its rank r is less than n. Thus, in the real field, a negative semi-definite quadratic form can be reduced to -y₁² - y₂² - ... - y_r² where r < n.. Its value q is less than or equal to zero for any nontrivial value of X .

Principal minor. A minor of a matrix A is called a principal minor if it is obtained by deleting selected (i.e. arbitrarily chosen) rows and the same numbered columns of A. Thus, the diagonal elements of a principal minor are diagonal elements of the matrix.

Theorem. Every symmetric matrix of rank r has at least one principal minor of order r different from zero.

Definite and semi-definite matrices. The matrix A of a real quadratic form

q = X^TAX is called definite or semi-definite according as the quadratic form is definite or semi-definite.

Theorems

1] A real symmetric matrix A is positive definite if and only if there exists a nonsingular matrix C such that A = C^TC.

2] A real symmetric matrix A of rank r is positive semi-definite if and only if there exists a matrix C of rank r such that A = C^TC..

3] If A is positive definite, every principal minor of A is positive.

4] If A is positive semi-definite, every principal minor of A is non-negative.

Leading principal minors of a symmetric matrix. The leading principal minors of a symmetric matrix

are defined as

5] Any n-square nonsingular matrix A can be rearranged by interchanging certain rows and the corresponding columns so that not both p_n-1and p_n-2 are zero.

6] If A is a symmetric matrix and if p_n-2p_n ≠ 0 but p_n-1= 0, then p_n-2 and p_n have opposite signs.

Regularly arranged symmetric matrix. A symmetric matrix is said to be regularly arranged if no two consecutive p’s in the sequence p₀, p₁, .... ,p_r are zero.

Regular quadratic form. A regular quadratic form is one whose matrix is regularly arranged.

7] Any symmetric matrix (quadratic form) of rank r can be regularly arranged.

8] A real quadratic form X^TAX is positive definite if and only if its rank is n and all leading principal minors are positive.

9] A real quadratic form X^TAX of rank r is positive semi-definite if and only if each of the leading principal minors p₀, p₁, .... ,p_r is positive.

References.

Ayres. Matrices (Schaum).