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B. Smith McMillan Forms

B.2 Polynomial Matrices

Multivariable transfer functions depend on polynomial matrices. There are a number of related terms that are used. Some of these are introduced here:

Definition B.1 A matrix is a polynomial matrix  if is a polynomial in , for and .

Definition B.2 A polynomial matrix is said to be a unimodular matrix  if its determinant is a constant. Clearly, the inverse of a unimodular matrix is also a unimodular matrix.

Definition B.3 An elementary operation on a polynomial matrix is one of the following three operations:

(eo1)
interchange of two rows or two columns;
(eo2)
multiplication of one row or one column by a constant;
(eo3)
addition of one row (column) to another row (column) times a polynomial.

Definition B.4 A left (right) elementary matrix is a matrix such that, when it multiplies from the left (right) a polynomial matrix, then it performs a row (column) elementary operation on the polynomial matrix. All elementary matrices are unimodular.

Definition B.5 Two polynomial matrices and are equivalent matrices, if there exist sets of left and right elementary matrices, and , respectively, such that (B.2.1)

Definition B.6 The rank of a polynomial matrix is the rank of the matrix almost everywhere in . The definition implies that the rank of a polynomial matrix is independent of the argument.

Definition B.7 Two polynomial matrices and having the same number of columns (rows) are right (left) coprime if all common right (left) factors are unimodular matrices.

Definition B.8 The degree ( ) of the column (row) ( ) of a polynomial matrix is the degree of highest power of in that column (row).

Definition B.9 A polynomial matrix is column proper if (B.2.2)

has a finite, nonzero value.

Definition B.10 A polynomial matrix is row proper if (B.2.3)

has a finite, nonzero value.