You are here : Control System Design  Index  Book Contents  Appendix B  Section B.2 B. Smith McMillan FormsB.2 Polynomial MatricesMultivariable transfer functions depend on polynomial matrices. There are a number of related terms that are used. Some of these are introduced here:
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:
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
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).
