Bilinear forms, Equivalence, Reduction to canonical form, Cogredient Transformations, Contragredient transformations

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Bilinear forms arise in various areas of mathematics and its applications. An example is in the problem of computing the correlation between variables in statistics.

Def. Homogeneous polynomial. A polynomial whose terms are all of the same degree with respect to all the variables taken together.

Example. x² + 3xy + 4y² is homogeneous.

Bilinear form. A bilinear form is a polynomial of the second degree which is linear and homogeneous in the two sets of variables (x₁, x₂, ... ,x_m ) and (y₁, y₂, ... ,y_m ) .

Example. f (x,y) = x₁y₁ + 5x₁y₂ - 2x₁y₃ + x₂y₁ - 4x₂y₃ is a bilinear form in the variables (x₁, x₂) and (y₁, y₂, y₃).

The most general bilinear form in the variables (x₁, x₂, ... ,x_m ) and (y₁, y₂, ... ,y_m ) can be written as

= X^TAY

where

and

The matrix of the coefficients is called the matrix of the bilinear form and the rank of A is called the rank of the form.

Change of variables in bilinear forms. Frequently it is necessary or desirable to introduce new variables into a bilinear form in place of X and Y. Let X^TAY be a bilinear form over a field F. Let X = BU and Y = CV be linear transformations relating X and Y to the variables U and V where the matrices B and C are also over F. Then

X^TAY = (BU)^TACV = U^T(B^TAC)V

Note. The matrices B and C may be either singular or nonsingular. No requirement is made in this regard. In either case the transformation carries matrix A over into matrix B^TAC. However, matrix B^TAC will be equivalent to A only if and only if the matrices B and C are nonsingular.

Equivalence of bilinear forms. Two bilinear forms are said to be equivalent over F if and only if there exist non-singular transformations X = BU and Y = CV over F which transform the first form into the second.

Non-singular linear transforms over a field F carry a bilinear form Q over F into another bilinear form which has the same rank as Q and is also over F.

Two bilinear forms with mxn matrices A and B over F are equivalent over F if and only if they have the same rank.

Reduction to canonical form. If the rank of a bilinear form X^TAY is r, there exist non-singular matrices P and Q such that

Using a change of variables given by X = P^TU and Y = QV the bilinear form X^TAY is reduced to

Theorem. Any bilinear form over F of rank r can be reduced by non-singular linear transformations over F to the canonical form u₁v₁ + u₂v₂ + .... + u_rv_r.

Types of bilinear forms. A bilinear form X^TAY is called

There are two types of linear transformations of special interest in connection with bilinear forms – cogredient transformations and contragredient transformatrions.

Cogredient Transformations. When a bilinear form X^TAY has a matrix A that is n-square so that both X and Y are n-vectors we sometimes wish to subject both X and Y to the same transformation X = CU and Y = CV. This is called a cogredient transformation and the variables are said to have been transformed cogrediently. The effect of such a cogredient transformation is to take the form X^TAY into the form U^T(C^TAC)V. The matrix C of the transformation may be either singular or nonsingular. No requirement is made in this regard. In either case the transformation carries matrix A over into matrix C^TAC. However, matrix C^TAC will be congruent to A if and only if the matrix C is non-singular.

Theorems.

1] Two bilinear forms over F are equivalent under cogredient transformations of the variables if and only if their matrices are congruent over F.

2] A symmetric bilinear form remains symmetric under cogredient traansformations of the variables.

3] A symmetric bilinear form of rank r can be reduced by nonsingular cogredient transformations of the variables to

4] A real symmetric bilinear form of rank r can be reduced by nonsingular cogredient transformations of the variables in the real field to

and in the complex field to

Contragredient transformations. Suppose a bilinear form X^TAY has a matrix A that is n-square so that both X and Y are n-vectors. Let us now subject this form to the linear transformation X = (C ^-1)^TU and Y = CV . This is called a contragredient transformation and the variables are said to have been transformed contragrediently. The effect of such a transformation is to take the form X^TAY into the form U^T(C ^-1AC)V. The importance of the contragredient transformation lies in the following theorem:

Theorem. The bilinear form

(where I_n is the identity matrix) is transformed into itself if and only if the two sets of variables are transformed contragrediently.

Factorable bilinear forms. A non-zero bilinear form is factorable if and only if its rank is one.

References.

Ayres. Matrices (Schaum).