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Consider two column matrices and . Suppose they are related to each other by means of a function so that

(3.168)

Depending on the mathematical features of the function , the relationship described by Eq. (3.168) is characterized by various designations, which are explained below.

1 (a) Homogeneous Versus Nonhomogeneous Relationships

The relationship set up by is called homogeneous if

(3.169)

It is called nonhomogeneous if

(3.170)

1 (b) Linear Versus Nonlinear Relationships

The relationship set up by is called linear if, for a scalar k and for all ,

(3.171)

It is called nonlinear if

(3.172)

In the case of a linear relationship, is expressed as follows in terms of an m × n matrix , which does not depend on :

(3.173)

Note that a linear relationship is also homogeneous, but a nonlinear relationship may or may not be homogeneous.

1 (c) Affine Relationship

The relationship set up by is called affine, if is expressed as follows in terms of an m × n matrix and an m × 1 matrix , which are both independent of .

(3.174)

In Eq. (3.174), is defined as the bias term. It may or may not be zero, i.e. .

Note that a general affine relationship with is both nonhomogeneous and nonlinear. However, a special affine relationship with happens to be both homogeneous and linear.