mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a polynomial matrix or matrix of polynomials is a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

whose elements are univariate or multivariate

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

s. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices. A univariate polynomial matrix ''P'' of degree ''p'' is defined as: :

P = \sum_^p A(n)x^n = A(0)+A(1)x+A(2)x^2+ \cdots +A(p)x^p

where

A(i)

denotes a matrix of constant coefficients, and

A(p)

is non-zero. An example 3×3 polynomial matrix, degree 2: :

P=\begin
1 & x^2 & x \\
0 & 2x & 2 \\
3x+2 & x^2-1 & 0
\end
=\begin
1 & 0 & 0 \\
0 & 0 & 2 \\
2 & -1 & 0
\end

+\begin
0 & 0 & 1 \\
0 & 2 & 0 \\
3 & 0 & 0
\endx+\begin
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 1 & 0
\endx^2.

We can express this by saying that for a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

''R'', the rings

M_n(R

and

(M_n(R)) /math> are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

Properties

*A polynomial matrix over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

with

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function. *The roots of a polynomial matrix over the

complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

are the points in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

where the matrix loses

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...

. *The determinant of a matrix polynomial with

Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...

positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...

(semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients. Note that polynomial matrices are ''not'' to be confused with

monomial matrices In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...

, which are simply matrices with exactly one non-zero entry in each row and column. If by λ we denote any element of the

over which we constructed the matrix, by ''I'' the identity matrix, and we let ''A'' be a polynomial matrix, then the matrix λ''I'' − ''A'' is the characteristic matrix of the matrix ''A''. Its determinant, , λ''I'' − ''A'', is the

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...

of the matrix ''A''.

References

* E.V.Krishnamurthy, Error-free Polynomial Matrix computations, Springer Verlag, New York, 1985 Matrices Polynomials {{Linear-algebra-stub