Polynomial and polynomial matrix glossary

Polynomial matrices

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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

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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 rank.

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

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Let the elements of the k×m polynomial matrix P be Then the numbers are the row and the column degrees of P, respectively.

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Suppose that P has column and row degrees respectively.

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

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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

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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 Roots

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The roots 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

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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.
Matrix pencil

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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

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There are three basic elementary row operations:
• multiplying a row by a nonzero constant, such as • interchanging two rows, such as • adding a polynomial multiple of one row to another, such as Elementary column operations are defined analogously.

Diophantine equations

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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

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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

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Theoretically, the degree of a polynomial 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

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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

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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 multiple of A and B then G is a least common right multiple of A and B.

Right divisors, left multiples, common right divisors, greatest common right divisors, common left multiples, and least common left multiples are similarly defined.

 Index Bézout equations coefficient matrix column degrees column leading coefficient matrix column rank column reduced conjugate coprime degree diagonal leading coefficient matrix diagonally reduced Diophantine equations divisor elementary row and column operations full normal rank full rank half diagonal degrees left coprime row reduced left prime matrix pencil monic multiple nonsingular normal rank para-Hermitian prime rank right coprime roots row degrees row leading coefficient matrix row rank Sylvester resultant matrix tall unimodular zeroing Top