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

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 .

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).

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