Polynomial and polynomial matrix glossary
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|Polynomial matrices||We review some definitions and
basic facts related to polynomial matrices.
A k×m polynomial matrix is a matrix of the form
where s is an indefinite variable (usually taking its values in the complex plane), and the k×m constant matrices
coefficient matrices. Usually, unless stated otherwise, we deal with real polynomial matrices, whose coefficient matrices are real.
If is not the zero matrix then we say that P has degree n. If is the unit matrix then P is said to be monic.
|Tall and wide||A polynomial or other matrix is tall if it has at least as many rows as columns. It is wide if it has at least as many columns as rows.|
|Rank||A polynomial matrix P
has full column rank (or full normal column rank) if it has full column rank
everywhere in the complex plane except at a finite number of points. Similar definitions
hold for full row rank and full
The normal rank of a polynomial matrix P equals
Similar definitions apply to the notions of normal column rank and normal row rank.
A square polynomial matrix is nonsingular if it has full normal rank.
|Row and column degrees||Let the elements of the k×m
polynomial matrix P be
Then the numbers
are the row and the column degrees of P, respectively.
|Leading coefficient matrices||Suppose that P has
column and row degrees
The column leading coefficient matrix of P is the constant matrix whose (i, j) entry is the coefficient of the term with power of the (i, j) entry of P.
The row leading coefficient matrix of P is the constant matrix whose (i, j) entry is the coefficient of the term with power of the (i, j) entry of P.
|Column and row reduced||A polynomial matrix is column reduced if its column leading coefficient matrix has full column rank. It is row reduced if its row leading coefficient matrix has full row rank.|
|Conjugate||If P is a polynomial
matrix then its conjugate P* is the polynomial
matrix defined by
The superscript H indicates the complex conjugate transpose.
|Para-Hermitian||A square polynomial matrix P is para-Hermitian if P* = P.|
|Diagonally reduced||The m×m
para-Hermitian polynomial matrix P is diagonally
reduced if there exist half diagonal degrees
so that the diagonal leading coefficient matrix
exists and is nonsingular. D is the diagonal polynomial matrix
or zeros of a polynomial matrix P are those points
in the complex plane where P loses rank.
If P is square then its roots are the roots of its determinant det P, including multiplicity.
|Primeness||A polynomial matrix P is left prime if it has full row rank everywhere in the complex plane. It is right prime if it has full column rank everywhere in the complex plane.|
|Coprimeness||The N polynomial
matrices with the same numbers of rows
are left coprime if
is left prime. If the N polynomila matrices all have the same numbers of columns then they are right coprime if
is right prime.
|Unimodular||A square polynomial matrix U is unimodular if its determinant det U is a nonzero constant. The inverse of a unimodular polynomial matrix is again a polynomial matrix.|
pencils are matrix polynomials of degree 1, such as
Matrix pencils are often represented as polynomial matrices of the special form
but we shall normally consider matrix pencils as general polynomial matrices of degree 1.
|Elementary row and column operations||There are three basic
elementary row operations:
Elementary column operations are defined analogously.
|Diophantine equations||The simplest type of linear
scalar polynomial equation - called Diophantine equation after by the Alexandrian
mathematician Diophantos (A.D. 275) is
The polynomials polynomials a, b and c are given while the polynomials x and y are unknown. The equation is solvable if and only if the greatest common divisor of a and b divides c. This implies that with relatively a and b coprime the equation is solvable for any right hand side polynomial, including c = 1.
The Diophantine equation possesses infinitely many solutions whenever it is solvable. If is any (particular) solution, then the general solution is
where t is an arbitrary polynomial (the parameter) and are coprime polynomials such that
If the a and b themselves are coprime then one can naturally take
Among all the solutions of Diophantine equation there exists a unique solution pair (x, y) characterized by
There is another (generally different) solution pair characterized by
The two solution pairs coincide only if
|Bézout equations||A Diophantine equation with 1
on its right hand side is called a Bézout equation. It may look like
with a and b given polynomials and x and y unknown.
|Zeroing||Theoretically, the degree of a
is n whenever . In numerical computations, however, one often encounters the case of very small (much smaller than the other coefficients) yet non-zero.
By way of example, consider two simple polynomials
where is almost (but not quit) zero. When computing the difference
a question on its degree may arise. It is necessary to compare with the norms of the other coefficients to decide whether or not the corresponding term should be completely deleted. This process is called zeroing. The performance of many algorithms for polynomial problems critically depends on the way zeroing is done, in particular when elementary operations are used.
|Sylvester resultant matrix||The Sylvester resultant matrix
corresponding to the polynomials
is the (m+n)×(m+n) constant matrix
The resultant matrix is nonsingular if and only if the polynomials a and b are coprime.
|Divisors and multiples||Consider polynomials a,
b and c such that a = bc. We say that b is a divisor
of a of or a is a multiple of b, and write b|a. This
is sometimes also stated as b divides a.
If a polynomial g divides both a and b then g is called a common divisor of a and b. If, furthermore, g is a multiple of every common divisor of a and b then g is a greatest common divisor of a and b. If the only common divisors of a and b are constants then the polynomials a and b are coprime.
If a polynomial m is a multiple of both a and b then m is called a common multiple of a and b. If, furthermore, m is a divisor of every common multiple of a and b then it is a least common multiple of a and b.
Next consider now polynomial matrices A, B and C of compatible sizes such that A = BC. We say that B is a left divisor of A or A is a right multiple of B.
If a polynomial matrix G is a left divisor of both A and B then G is called a common left divisor of A and B. If, furthermore, G is a right multiple of every common left divisor of A and B then it is a greatest common left divisor of A and B. If the only common left divisors of A and B are unimodular matrices then the polynomial matrices A and B are left coprime.
If a polynomial matrix M is a right multiple of
both A and B then M is called a common right multiple of A
and B. If, furthermore, L is a left divisor of every common
Right divisors, left multiples, common right divisors, greatest common right divisors, common left multiples, and least common left multiples are similarly defined.